p.34
Poisson Distribution: Definition and Applications
What are the key characteristics of a Poisson distribution?
It describes events that occur independently and at a constant average rate.
p.41
Poisson Distribution: Definition and Applications
What is the probability distribution for an area of 100 cm²?
X ~ Poisson(μ = 2) and Y ~ Poisson(μ = 4).
p.8
Cumulative Distribution Function (CDF)
What is a key property of the CDF?
F(x) is an increasing function from 0 to 1.
p.53
Comparison of Binomial and Poisson Distributions
How do the draws differ when sampling with replacement versus without replacement?
With replacement, the draws are independent; without replacement, the draws are not independent.
p.56
Applications of Discrete Probability Distributions in Real Life
What does the variable 'x' represent in the context of the problem?
The number of tagged fishes found in the recapture (16).
p.26
Binomial Distribution: Definition and Examples
What does the notation [n choose x] represent in the Binomial Distribution?
[n choose x] represents the number of ways to choose x successes in n trials, calculated as n! / [x!(n-x)!].
p.16
Discrete Random Variables: Definitions and Properties
What is the relationship between the random variables Y and X when Y = aX + c?
Y is also a discrete random variable.
p.21
Binomial Distribution: Definition and Examples
What is the binomial distribution used for in the given example?
To find the probability of obtaining 0, 1, and 2 heads when tossing a fair coin twice.
p.46
Comparison of Binomial and Poisson Distributions
If X ~ B(i, p), what does X approximate when the conditions are met?
X can be well-approximated by Y ~ P(μ), where μ = np.
p.38
Comparison of Binomial and Poisson Distributions
What is the approximate value of e in the limit expression lim n→∞ (1 + 1/n)^n?
The approximate value is 2.718.
p.45
Comparison of Binomial and Poisson Distributions
When can the Poisson approximation be applied to the Binomial distribution?
When the number of trials is large and the probability of success is small.
p.54
Hypergeometric Distribution: Definition and Examples
What conclusion can the buyer draw from the probability calculation?
It can give him confidence in accepting the lot.
p.2
Binomial Distribution: Definition and Examples
What is an example of a discrete probability distribution involving coin flips?
Flipping a coin 50 times and observing the number of heads.
p.13
Expected Value and Variance of Discrete Random Variables
What is the alternative formula for calculating variance used in the example?
σ² = Σ [ (x - μ)² * Pr(X=x) ]
p.7
Cumulative Distribution Function (CDF)
What is a key property of the CDF?
It is an increasing function from 0 to 1.
p.29
Binomial Distribution: Definition and Examples
How is the maximum probability value for Pr(X = k) determined?
By comparing Pr(X = w + 1) and Pr(X = w).
p.22
Binomial Distribution: Definition and Examples
What happens to the proportion of neutrophils in patients with bacterial or acute viral infections?
It is much higher than 70%.
p.43
Poisson Distribution: Definition and Applications
What is the goal regarding the probability of chocolate drops in cookies?
To ensure that the probability of each cookie containing at least 1 chocolate drop is greater than 99%.
p.38
Comparison of Binomial and Poisson Distributions
What does the expression k! in the limit formula signify?
k! is the factorial of k, representing the number of ways to arrange k successes.
p.4
Discrete Random Variables: Definitions and Properties
What does the random variable X represent in the context of weight?
X = weight of an individual.
p.35
Poisson Distribution: Definition and Applications
What kind of events does the Poisson Distribution describe?
Independent events occurring over a period of time or space.
p.54
Hypergeometric Distribution: Definition and Examples
What is the scenario described in the example?
A buyer checks a sample of machine parts from a lot to determine if he should accept it.
p.47
Binomial Distribution: Definition and Examples
What does the 47 Rule of Thumb state about the approximation of random variables?
Pr(X = x) ≈ Pr(Y = x) for all possible values of x.
p.54
Hypergeometric Distribution: Definition and Examples
What distribution does the number of defectives found in the sample follow?
Hypergeometric distribution.
p.41
Poisson Distribution: Definition and Applications
How does the shape of the Poisson distribution change with small μ?
It is heavily skewed to the right.
p.26
Binomial Distribution: Definition and Examples
How is the expected number of successes in n trials calculated?
It is calculated by multiplying the probability of success in one trial (p) by the number of trials (n).
p.60
Negative Binomial Distribution: Definition and Examples
What does the Negative Binomial Distribution model?
The number of times a fixed number of successes occurs in a series of experiments.
p.7
Cumulative Distribution Function (CDF)
What happens to the CDF as the number of possible values increases?
The CDF will get smoother and smoother.
p.4
Discrete Random Variables: Definitions and Properties
What is an example of a random variable when tossing two dice?
X = Sum of the numbers on the two dice.
p.27
Expected Value and Variance of Discrete Random Variables
What is the formula for E[X²] in a binomial distribution?
E[X²] = Σ (x² * P(X=x)) for x = 0 to n.
p.24
Binomial Distribution: Definition and Examples
What is the shape of the binomial distribution when p = 0.5?
The distribution is symmetric.
p.3
Discrete Random Variables: Definitions and Properties
Give an example of a discrete random variable.
The number of heads in a series of coin flips.
p.42
Poisson Distribution: Definition and Applications
How can Poisson probability be calculated rapidly?
By using the recursion relation: Pr(X = x) = μ^x * Pr(X = x - 1), for x = 1, 2, ...
p.5
Discrete Random Variables: Definitions and Properties
What is a random variable (r.v.)?
A numeric quantity that takes different values with specified probabilities.
p.56
Applications of Discrete Probability Distributions in Real Life
What does the variable 'i' represent in the context of the problem?
The number of fishes recaptured (150).
p.55
Hypergeometric Distribution: Definition and Examples
What distribution does the number of recaptured tagged fishes follow?
Hypergeometric distribution.
p.50
Poisson Distribution: Definition and Applications
What is the goal for the number of spare calculators to bring for the exam?
To ensure all students have calculators with at least a 90% chance.
p.19
Binomial Distribution: Definition and Examples
What are the parameters of a Binomial distribution?
The number of trials (n) and the probability of success (p).
p.7
Cumulative Distribution Function (CDF)
What is the significance of the empty circle in the CDF?
It indicates that F(1) is excluded from the value 0.129.
p.19
Binomial Distribution: Definition and Examples
How is a random variable X defined in the context of Binomial distribution?
X is the total number of successes in n independent experiments.
p.58
Geometric and Negative Binomial Distributions
What does Pr(X > k) represent in the context of the Geometric Distribution?
It represents the probability of needing more than k trials to achieve the first success.
p.25
Binomial Distribution: Definition and Examples
What is the mean of a Binomial Distribution?
μ = E[X] = np, where p is the probability of success.
p.3
Discrete Random Variables: Definitions and Properties
Can discrete random variables take on any value within a range?
No, they can only take specific, distinct values.
p.10
Expected Value and Variance of Discrete Random Variables
What are the measures of location for a sample?
Sample Mean, Median, Mode.
p.53
Hypergeometric Distribution: Definition and Examples
What is the distribution of the number of white balls drawn without replacement from the same box?
X follows a hypergeometric distribution with N = 30, N1 = 10, N2 = 20, i = 5.
p.2
Binomial Distribution: Definition and Examples
In the coin flip example, what are we calculating?
The probabilities of obtaining 0, 1, 2, ..., 50 heads.
p.18
Binomial Distribution: Definition and Examples
What is a Binomial Distribution?
A probability distribution that summarizes the likelihood of a value taking on two independent values under a given set of parameters.
p.58
Geometric and Negative Binomial Distributions
What does the Memorylessness property imply about trials in an experiment?
The conditional probability distribution of additional trials does not depend on how many failures have been observed.
p.19
Binomial Distribution: Definition and Examples
What does the Binomial distribution represent?
The probability distribution on the number of successes in independent experiments.
p.33
Binomial Distribution: Definition and Examples
What is the distribution of Y, the number of pairs of twins born?
Y follows a Binomial distribution with parameters n = 120 and p = 0.016.
p.25
Expected Value and Variance of Discrete Random Variables
How is the variance of a Binomial Distribution calculated?
σ² = np(1 - p), where n is the number of trials and p is the probability of success.
p.4
Discrete Random Variables: Definitions and Properties
What is a random variable (r.v.)?
A numeric quantity that takes different values with specified probabilities.
p.46
Poisson Distribution: Definition and Applications
What is the condition for p in a Poisson distribution?
p should be small (< 0.11).
p.30
Binomial Distribution: Definition and Examples
What is the conditional probability of getting 6 heads given that there are at least 4 heads when tossing a fair coin 5 times?
Pr(X = 5 | X ≥ 4) = Pr(X = 5) / (Pr(X = 4) + Pr(X = 5)) = 1/6.
p.4
Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
What does P(X = x) represent?
The probability that the random variable X takes the value x.
p.29
Binomial Distribution: Definition and Examples
What indicates that the probability is monotonically increasing?
If Pr(X = t + 1) / Pr(X = t) ≥ 1.
p.36
Comparison of Binomial and Poisson Distributions
What is a key characteristic of the Binomial Distribution?
It has a finite number of trials.
p.37
Binomial Distribution: Definition and Examples
What is the formula for the probability of getting exactly k successes in n trials in a Binomial distribution?
The formula is P(X = k) = (n choose k) * p^k * (1 - p)^(n - k).
p.36
Comparison of Binomial and Poisson Distributions
What is a key characteristic of the Poisson Distribution?
The number of trials can be infinite.
p.15
Discrete Random Variables: Definitions and Properties
What is the relationship between the random variables Y and X when Y = aX + c?
Y is also a discrete random variable.
p.19
Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
What does the term 'n choose x' refer to in the pmf of the Binomial distribution?
It refers to the number of ways to choose x successes from n trials.
p.19
Binomial Distribution: Definition and Examples
What are the possible values of x in a Binomial distribution?
x can take values from 0 to n.
p.34
Poisson Distribution: Definition and Applications
In what scenarios is the Poisson distribution typically applied?
In scenarios like the number of phone calls received at a call center in an hour.
p.26
Binomial Distribution: Definition and Examples
What is the mean of a Binomial Distribution?
The mean (μ) is calculated as E(X) = np, where n is the number of trials and p is the probability of success.
p.5
Discrete Random Variables: Definitions and Properties
What defines a continuous random variable?
A random variable that can take a continuous range of values over an interval.
p.37
Comparison of Binomial and Poisson Distributions
What is the primary difference between Binomial and Poisson distributions?
Binomial distribution is used for a fixed number of trials with two outcomes, while Poisson distribution is used for counting the number of events in a fixed interval of time or space.
p.30
Binomial Distribution: Definition and Examples
What is the probability of rolling a 1 exactly twice when rolling an unbiased die 7 times?
Pr(X = 2) where X ~ Binomial(7, 1/6).
p.37
Binomial Distribution: Definition and Examples
In a Binomial distribution, what does 'p' represent?
'p' represents the probability of success in a single trial.
p.30
Binomial Distribution: Definition and Examples
What is the conditional probability of the first 2 rolls being a 1 given that a 1 appears exactly twice in 7 rolls?
Pr(Y = 0 | X = 2) where Y ~ Binomial(5, 1/6).
p.22
Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
What is the probability mass function (pmf) for the number of neutrophils in a sample of 10?
Pr(X = x) = (10! / (x! (10 - x)!)) * (0.7^x) * (0.3^(10-x)), where x = 0, 1, ..., 10.
p.30
Binomial Distribution: Definition and Examples
What does the variable X represent in the context of tossing a fair coin?
The number of heads among 5 tosses.
p.22
Binomial Distribution: Definition and Examples
What is the probability of observing 10 neutrophils in the sample?
Pr(X = 10) = 0.7^10 = 0.0282.
p.7
Cumulative Distribution Function (CDF)
How is the CDF graphically illustrated?
It looks like a series of steps, called the step function.
p.6
Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
How does a probability mass function relate to a frequency distribution?
PMF specifies the population, while frequency distribution summarizes the sample.
p.24
Binomial Distribution: Definition and Examples
What is the shape of the binomial distribution when p > 0.5?
The distribution is left-skewed.
p.34
Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
How is the probability mass function (PMF) of a Poisson distribution defined?
P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of events.
p.45
Comparison of Binomial and Poisson Distributions
What is the Poisson approximation used for?
To approximate the Binomial distribution under certain conditions.
p.54
Hypergeometric Distribution: Definition and Examples
What happens to the probability of finding 0 defectives if the number of defectives in the lot increases?
The probability will be even lower.
p.53
Comparison of Binomial and Poisson Distributions
What is the probability of drawing a white ball from the box?
1/3 (since there are 10 white balls out of 30 total).
p.55
Hypergeometric Distribution: Definition and Examples
What is the formula to calculate the probability of recapturing 16 tagged fishes?
Pr(X = 16) = (N1 choose 16) * ((N - N1) choose (n - 16)) / (N choose n).
p.46
Comparison of Binomial and Poisson Distributions
What are the three conditions for approximating a binomial distribution with a Poisson distribution?
i) n ≥ 2000, ii) p < 0.11, iii) np < 5.
p.60
Negative Binomial Distribution: Definition and Examples
What is the probability formula for k successes in the Negative Binomial Distribution?
Pr(X = k) = (k - 1) choose (V - 1) * (1 - p)^(k - V) * p^V.
p.4
Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
What is the probability notation for a range of values for a random variable?
P(a < X < b) represents the probability that X falls between a and b.
p.44
Poisson Distribution: Definition and Applications
How do you calculate the probability of at least 2 calls in the next minute?
Pr(X ≥ 2) = 1 - Pr(X = 0) - Pr(X = 1).
p.2
Poisson Distribution: Definition and Applications
What is the probability we are calculating for the soccer match example?
The probability that the actual scores are 1:0.
p.32
Binomial Distribution: Definition and Examples
What is the probability that the total number of heads (W) equals 0?
Pr(X + Y = 0) = Pr(X = 0) * Pr(Y = 0) = 0.5^5
p.37
Poisson Distribution: Definition and Applications
What is the mean of a Poisson distribution?
The mean of a Poisson distribution is equal to its parameter λ (lambda).
p.33
Comparison of Binomial and Poisson Distributions
How do you compute Pr(μ - σ ≤ Y ≤ μ + σ)?
Use the normal approximation to the binomial distribution for large n.
p.33
Comparison of Binomial and Poisson Distributions
What approximation can be used for a binomial distribution with large n and small p?
It can be approximated by a Poisson distribution.
p.34
Poisson Distribution: Definition and Applications
What is the parameter of the Poisson distribution?
The average number of events (λ) in the given interval.
p.54
Hypergeometric Distribution: Definition and Examples
What is the assumption about the number of defective items in the lot?
There are actually 6 defectives in the lot.
p.5
Discrete Random Variables: Definitions and Properties
What notation is commonly used for random variables?
Capital letters such as X, Y, Z.
p.11
Expected Value and Variance of Discrete Random Variables
What do x₁, x₂, ..., xᵢ represent in the variance formula?
They are the possible values of the random variable X.
p.4
Discrete Random Variables: Definitions and Properties
What notation is commonly used for random variables?
Capital letters such as X, Y, Z.
p.46
Expected Value and Variance of Discrete Random Variables
What does E(X) = np < 5 indicate in the context of Poisson distribution?
'Rare events' means only a few events are expected to occur.
p.26
Binomial Distribution: Definition and Examples
What is the extension of the Binomial Distribution formula?
The extension involves calculating E(X) using the formula E(X) = np, where p is the probability of success.
p.58
Geometric and Negative Binomial Distributions
What is the formula for the cumulative distribution function (CDF) of a Geometric Distribution?
F_X(x) = Pr(X ≤ x) = 1 - (1 - p)^x.
p.21
Binomial Distribution: Definition and Examples
What are the parameters of the binomial distribution in this example?
i = 2 (number of trials), p = 0.5 (probability of heads).
p.7
Cumulative Distribution Function (CDF)
What is the value of F(0.99999) and F(1.00000)?
F(0.99999) = 0.129 and F(1.00000) = 0.393.
p.36
Comparison of Binomial and Poisson Distributions
How can the Poisson Distribution be conceptualized in relation to the Binomial Distribution?
As a Binomial distribution with an infinite number of experiments.
p.24
Binomial Distribution: Definition and Examples
What is the binomial distribution with n = 10?
It refers to a binomial distribution with 10 trials.
p.10
Expected Value and Variance of Discrete Random Variables
How is the expected value (μ) mathematically represented?
E(X) = μ = Σ (x_i * Pr(X = x_i)) for i = 1 to n.
p.11
Expected Value and Variance of Discrete Random Variables
What is the alternative formula for population variance?
σ² = E[X²] - (E[X])² = Σ (xᵢ² Pr(X = xᵢ)) - (Σ (xᵢ Pr(X = xᵢ)))²
p.13
Expected Value and Variance of Discrete Random Variables
What are the probabilities associated with the outcomes for X?
Pr(X=0) = 1/4, Pr(X=1) = 1/2, Pr(X=2) = 1/4.
p.45
Comparison of Binomial and Poisson Distributions
How does the mean of the Poisson distribution relate to the Binomial distribution?
The mean of the Poisson distribution is equal to np, where n is the number of trials and p is the probability of success.
p.49
Binomial Distribution: Definition and Examples
What distribution does the number of students without a calculator follow?
Binomial distribution, specifically B(100, 0.02).
p.2
Poisson Distribution: Definition and Applications
What are we calculating in the paper production example?
The probabilities of obtaining 0, 1, 2,... holes.
p.58
Geometric and Negative Binomial Distributions
How is the probability of X being greater than k + j expressed?
Pr(X > k + j) = (1 - p)^(k+j).
p.21
Binomial Distribution: Definition and Examples
What is the formula for calculating the probability of obtaining x heads?
Pr(X = x) = i! / (x! (i - x)!) * p^x * (1 - p)^(i - x).
p.44
Poisson Distribution: Definition and Applications
How do you calculate the probability of at least 2 calls in the next 6 minutes?
Pr(Y ≥ 2) = 1 - Pr(Y = 0) - Pr(Y = 1).
p.6
Discrete Random Variables: Definitions and Properties
In the example of tossing two fair coins, what does the random variable X represent?
The number of heads observed.
p.40
Poisson Distribution: Definition and Applications
What is the probability of finding 5 or more colonies in 200 cm²?
Pr(Y ≥ 5) = 1 - Pr(Y ≤ 4) = 0.3711.
p.34
Poisson Distribution: Definition and Applications
What is the Poisson distribution used for?
To model the number of events occurring in a fixed interval of time or space.
p.25
Expected Value and Variance of Discrete Random Variables
What happens to the variance when p = 0 or p = 1?
Variance σ² = 0 due to no uncertainty on X.
p.53
Comparison of Binomial and Poisson Distributions
What is the distribution of the number of white balls drawn with replacement from a box of 10 white and 20 black balls?
X follows a Binomial distribution: X ∼ B(5, 1/3).
p.13
Expected Value and Variance of Discrete Random Variables
What is the intuition behind the expected value of 1 when tossing 2 fair coins?
We expect to obtain 1.0 head.
p.46
Poisson Distribution: Definition and Applications
What is the requirement for the number of experiments to observe 'rare events' in a Poisson distribution?
A large n (≥ 2000) is needed.
p.26
Binomial Distribution: Definition and Examples
What is the formula for the expected value E(X) in a Binomial Distribution?
E(X) = Σ (from x=0 to n) [n choose x] p^x (1-p)^(n-x).
p.18
Binomial Distribution: Definition and Examples
What parameters define a Binomial Distribution?
The number of trials (n) and the probability of success (p).
p.38
Comparison of Binomial and Poisson Distributions
What is the significance of the parameter μ in the context of the Poisson distribution?
μ represents the mean number of occurrences in a fixed interval for the Poisson distribution.
p.27
Expected Value and Variance of Discrete Random Variables
What is the relationship between E[X²] and variance in a binomial distribution?
Var(X) = E[X²] - (E[X])².
p.36
Comparison of Binomial and Poisson Distributions
In the Poisson Distribution, what happens to the probability as the number of events increases?
The probability becomes very small when the number of events gets larger.
p.19
Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
What is the probability mass function (pmf) of the Binomial distribution?
Pr(X = x) = (n choose x) * p^x * (1 - p)^(n - x), for x = 0, 1, 2, ..., n.
p.35
Poisson Distribution: Definition and Applications
What is the probability formula for a Poisson random variable?
Pr(X = k) = (w^(-μ) * μ^k) / k!, where k = 0, 1, 2, ...
p.35
Poisson Distribution: Definition and Applications
What does the parameter μ represent in the Poisson Distribution?
The expected number of events to occur.
p.47
Binomial Distribution: Definition and Examples
What is the condition for X2 ~ Binomial(20, 0.1)?
It has marginal n and p, making it an okay approximation.
p.47
Binomial Distribution: Definition and Examples
What is the condition for X3 ~ Binomial(1000, 0.002)?
n is large and p is small, making it a good approximation.
p.45
Poisson Distribution: Definition and Applications
What is the parameter of the Poisson distribution?
The average rate (λ) of occurrence.
p.5
Discrete Random Variables: Definitions and Properties
What defines a discrete random variable?
A random variable for which there exists a discrete set of numeric values.
p.29
Binomial Distribution: Definition and Examples
What happens if ip is an integer in the context of the mode?
Then E[X] = ip is the mode.
p.40
Poisson Distribution: Definition and Applications
What is the expected number of bacterial colonies per 100 cm²?
4 (calculated as 0.02 * 100).
p.37
Poisson Distribution: Definition and Applications
What type of events does the Poisson distribution model?
It models the number of events occurring in a fixed interval of time or space.
p.30
Binomial Distribution: Definition and Examples
What does the variable Y represent in the context of rolling a die?
The number of times a 1 appears among the last 5 rolls.
p.38
Comparison of Binomial and Poisson Distributions
How does the Poisson distribution relate to the Binomial distribution when n is large and p is small?
The Poisson distribution serves as an approximation for the Binomial distribution under these conditions.
p.37
Comparison of Binomial and Poisson Distributions
When is it appropriate to use a Poisson distribution instead of a Binomial distribution?
When the number of trials is large and the probability of success is small.
p.15
Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
What does Pr(1.4 < X < 3.1) represent?
The probability that the random variable X falls between 1.4 and 3.1.
p.24
Binomial Distribution: Definition and Examples
What is the binomial distribution with n = 4?
It refers to a binomial distribution with 4 trials.
p.3
Discrete Random Variables: Definitions and Properties
What is the significance of discrete random variables in probability?
They are used to model scenarios where outcomes are distinct and countable.
p.41
Poisson Distribution: Definition and Applications
What shape does the Poisson distribution have?
It is a right-skewed distribution.
p.12
Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
What does PMF stand for in probability?
Probability Mass Function.
p.41
Poisson Distribution: Definition and Applications
What happens to the Poisson distribution as μ increases?
It becomes more symmetric, though still slightly right-skewed.
p.8
Cumulative Distribution Function (CDF)
In the example of tossing two fair coins, what does the random variable X represent?
The number of heads observed.
p.7
Cumulative Distribution Function (CDF)
What is the definition of the Cumulative Distribution Function (CDF) for a discrete random variable X at value x?
The probability that X is less than or equal to the value x (Notation: Pr(X ≤ x)).
p.60
Negative Binomial Distribution: Definition and Examples
What are the parameters of the Negative Binomial Distribution?
The number of successes (V) and the probability of success (p).
p.38
Comparison of Binomial and Poisson Distributions
What is the relationship between Binomial and Poisson distributions as n approaches infinity?
As n approaches infinity, the Binomial distribution can be approximated by the Poisson distribution under certain conditions.
p.55
Hypergeometric Distribution: Definition and Examples
What is the goal in estimating the total number of fishes (N)?
To choose N such that Pr(X = 16) is largest.
p.49
Comparison of Binomial and Poisson Distributions
What is the rule of thumb for approximating a Binomial distribution with a Poisson distribution?
i ≥ 20, p < 0.1, and ip = 2 < 5.
p.27
Binomial Distribution: Definition and Examples
What does the term 'p' represent in the binomial distribution?
The probability of success in a single trial.
p.40
Poisson Distribution: Definition and Applications
What is the probability of finding 4 colonies in 200 cm²?
Pr(Y = 4) = e^(-4) * (4^4 / 4!) = 0.1954.
p.1
Introduction to Discrete Probability Distributions
What is the focus of Chapter 3 in STAT 1012?
Discrete Probability Distributions.
p.24
Binomial Distribution: Definition and Examples
What happens to the binomial distribution when p < 0.5?
The distribution is right-skewed.
p.10
Expected Value and Variance of Discrete Random Variables
What is the expected value (population mean) of a random variable X?
The sum of the product of all possible values with their corresponding probabilities.
p.47
Binomial Distribution: Definition and Examples
What is the condition for X1 ~ Binomial(10, 0.2)?
n is too small and p is too large, making it a bad approximation.
p.8
Cumulative Distribution Function (CDF)
What is the Cumulative Distribution Function (CDF) of a discrete random variable X at value x?
The probability that X is less than or equal to the value x (Notation: Pr(X ≤ x)).
p.45
Binomial Distribution: Definition and Examples
What are the parameters of the Binomial distribution?
Number of trials (n) and probability of success (p).
p.29
Binomial Distribution: Definition and Examples
What is the mode of a Binomial distribution?
The mode (k) is the largest integer less than or equal to i + 1p.
p.53
Hypergeometric Distribution: Definition and Examples
What are the parameters for the hypergeometric distribution in this example?
N = 30 (total), N1 = 10 (white), N2 = 20 (black), i = 5 (draws).
p.4
Discrete Random Variables: Definitions and Properties
How is a random variable defined mathematically?
As a function from a sample space S into the real numbers.
p.29
Binomial Distribution: Definition and Examples
What indicates that the probability is monotonically decreasing?
If Pr(X = t + 1) / Pr(X = t) ≤ 1.
p.18
Binomial Distribution: Definition and Examples
What is the formula for the probability mass function (PMF) of a Binomial Distribution?
P(X = k) = (n choose k) * p^k * (1-p)^(n-k).
p.49
Poisson Distribution: Definition and Applications
How is the probability Pr(X ≤ 1) calculated using the Poisson approximation?
Pr(Y ≤ 1) = e^(-2) * (2^0 / 0!) + e^(-2) * (2^1 / 1!) = 3e^(-2).
p.18
Binomial Distribution: Definition and Examples
What does 'n choose k' represent in the Binomial Distribution formula?
The number of ways to choose k successes in n trials.
p.23
Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
What is the probability mass function (pmf) formula for the number of neutrophils?
Pr(X = x) = (10! / (x! (10 - x)!)) * (0.7^x) * (0.3^(10-x)), where x = 0, 1, ..., 10.
p.18
Expected Value and Variance of Discrete Random Variables
What is the variance of a Binomial Distribution?
Var(X) = n * p * (1 - p).
p.43
Poisson Distribution: Definition and Applications
What does Pr(X = 0) represent in this context?
The probability that there are no chocolate drops in a cookie.
p.21
Binomial Distribution: Definition and Examples
What does it mean when p = 0.5 in a binomial distribution?
The distribution is symmetric.
p.3
Discrete Random Variables: Definitions and Properties
What is a discrete random variable?
A variable that can take on a countable number of distinct values.
p.11
Expected Value and Variance of Discrete Random Variables
What is the formula for population variance (σ²)?
σ² = E[X - μ²] = Σ (xᵢ - μ)² Pr(X = xᵢ)
p.11
Expected Value and Variance of Discrete Random Variables
What does the variance measure in a population?
The sum of squares of all possible values of X minus the mean (μ) with their corresponding probabilities.
p.5
Discrete Random Variables: Definitions and Properties
What is the relationship between a sample space S and a random variable X?
A random variable is a function from a sample space S into the real numbers.
p.58
Geometric and Negative Binomial Distributions
What is the property of Memorylessness in probability?
Pr(X > k + j | X > k) = Pr(X > j), where k and j are non-negative integers.
p.5
Discrete Random Variables: Definitions and Properties
What are the two types of quantitative variables?
Discrete and continuous variables.
p.38
Comparison of Binomial and Poisson Distributions
What does the limit expression lim n→∞ (k/n) represent in the context of the Binomial distribution?
It represents the probability p in the Binomial distribution as n approaches infinity.
p.4
Discrete Random Variables: Definitions and Properties
What is an example of a random variable when tossing a coin 25 times?
X = Number of heads in 25 tosses.
p.6
Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
What is another name for the probability mass function?
Probability distribution.
p.16
Expected Value and Variance of Discrete Random Variables
What is the expectation of a linear combination of functions of X?
E(a w1(X) + b w2(X) + c) = aE(w1(X)) + bE(w2(X)) + c.
p.15
Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
What is the PMF of the new random variable W = X²?
It is derived from the probability distribution of X by squaring its values.
p.33
Expected Value and Variance of Discrete Random Variables
What is the expected value (μ) of Y?
μ = E(Y) = n * p = 120 * 0.016 = 1.92.
p.49
Binomial Distribution: Definition and Examples
What is the exact calculation for Pr(X ≤ 1) using the Binomial distribution?
Pr(X ≤ 1) = (0.98)^100 + 100(0.98)^99(0.02) = 0.4032.
p.36
Applications of Discrete Probability Distributions in Real Life
In astronomy, what does the Poisson Distribution model?
The number of photons arriving at a telescope.
p.33
Expected Value and Variance of Discrete Random Variables
What is the variance (σ²) of Y?
σ² = V(Y) = n * p * (1 - p) = 120 * 0.016 * (1 - 0.016) ≈ 1.89.
p.40
Poisson Distribution: Definition and Applications
What is the probability of finding 0 colonies in 200 cm²?
Pr(Y = 0) = e^(-4) = 0.0183.
p.36
Applications of Discrete Probability Distributions in Real Life
Give an example of a real-life application of the Poisson Distribution in electrical systems.
The number of telephone calls arriving.
p.40
Poisson Distribution: Definition and Applications
What is the probability of finding 3 colonies in 200 cm²?
Pr(Y = 3) = e^(-4) * (4^3 / 3!) = 0.1954.
p.15
Expected Value and Variance of Discrete Random Variables
How do you compute the variance of Y if Y = 1 - 4X?
Use the formula V(Y) = 16V(X).
p.22
Applications of Discrete Probability Distributions in Real Life
What could be the implications of observing all neutrophils in the sample?
The person is either healthy but unlucky or has a viral/bacterial infection (p >> 0.7).
p.27
Variance of Discrete Random Variables
What is the significance of the term '1 - p' in the variance formula?
It represents the probability of failure in a single trial.
p.23
Applications of Discrete Probability Distributions in Real Life
What could it indicate if a person has a high proportion of neutrophils?
The person may have a viral or bacterial infection.
p.43
Poisson Distribution: Definition and Applications
What inequality must hold for Pr(X = 0)?
Pr(X = 0) = e^(-μ) ≤ 0.01.
p.32
Binomial Distribution: Definition and Examples
How do you calculate the probability that W equals 1?
Pr(X + Y = 1) = Pr(X = 0, Y = 1) + Pr(X = 1, Y = 0) = 5 * 0.5^5
p.21
Binomial Distribution: Definition and Examples
How does the probability change as the number of trials increases?
The probabilities for k successes become more spread out and can be calculated using the binomial formula.
p.15
Cumulative Distribution Function (CDF)
What is the cumulative distribution function (CDF) of X?
It is a function that shows the probability that X will take a value less than or equal to x.
p.15
Expected Value and Variance of Discrete Random Variables
What is the first step in computing the expected value E(X)?
Identify the probability distribution of X.
p.23
Binomial Distribution: Definition and Examples
What does a probability of Pr(X = 0) = 0.0000059 suggest?
It suggests that having no neutrophils is extremely unlikely.
p.32
Binomial Distribution: Definition and Examples
What is the property of the sum of two independent binomial random variables?
If X ~ B(n1, p) and Y ~ B(n2, p), then X + Y ~ B(n1 + n2, p).
p.32
Binomial Distribution: Definition and Examples
What is the question regarding the difference of the two random variables (X - Y)?
The distribution of X - Y is not straightforward and requires further analysis.
p.36
Applications of Discrete Probability Distributions in Real Life
How can the Binomial Distribution be applied in a coin flipping scenario?
By flipping a coin n times.
p.40
Poisson Distribution: Definition and Applications
What is the probability of finding 1 colony in 200 cm²?
Pr(Y = 1) = e^(-4) * (4^1 / 1!) = 0.0733.
p.40
Poisson Distribution: Definition and Applications
What is the probability of finding 2 colonies in 200 cm²?
Pr(Y = 2) = e^(-4) * (4^2 / 2!) = 0.1465.
p.36
Applications of Discrete Probability Distributions in Real Life
How is the Poisson Distribution used in biology?
To model the number of mutations on DNA per unit time.
p.36
Applications of Discrete Probability Distributions in Real Life
In biology, how is the Poisson Distribution relevant to white blood cells?
It models the number of neutrophils out of n white blood cells.
p.36
Applications of Discrete Probability Distributions in Real Life
What is a financial application of the Poisson Distribution?
The number of losses or claims occurring in a given period of time.
