Discrete Probability Distributions: Master the Fundamentals for MSc Statistics

Question 1

A random variable X follows a hypergeometric distribution with parameters N = 10, K = 5, and n = 3. What is the probability of X taking the value 2?

Accepted Answer

0.2

Answer

0.1

Answer

0.3

Answer

0.4

Question 2

The hypergeometric distribution is used to calculate:

Accepted Answer

The probability of drawing a specific number of successes from a finite population without replacement

Answer

The probability of obtaining a specific sample mean

Answer

The probability of rolling a specific number on a dice

Answer

The probability of a given outcome in a lottery draw

Question 3

The mean of a binomial distribution with parameters n and p is:

Accepted Answer

np

Answer

np(1-p)

Answer

n(1-p)

Answer

p(1-p)

Question 4

The variance of a binomial distribution with parameters n and p is:

Accepted Answer

np(1-p)

Answer

n(1-p)

Answer

np

Answer

p(1-p)

Question 5

A call center receives an average of 5 calls per hour, following a Poisson distribution. Calculate the probability of receiving exactly 3 calls in an hour.

Accepted Answer

0.1404

Answer

0.2337

Answer

0.3557

Question 6

The probability of an event occurring is 0.4. Determine the probability of its non-occurrence.

Accepted Answer

0.6

Answer

0.4

Answer

1.0

Question 7

Which type of probability involves the likelihood of one event influencing the occurrence of another?

Accepted Answer

Dependent probability

Answer

Independent probability

Answer

Joint probability

Question 8

If the probability of event A is 0.3 and the probability of event B is 0.4, and A and B are independent events, calculate the probability of both events occurring.

Accepted Answer

0.12

Answer

0.70

Answer

1.00

Question 9

Which characteristic does NOT apply to the binomial distribution?

Accepted Answer

Symmetric shape

Answer

Discrete outcomes

Answer

Mean of np

Answer

Variance of np(1-p)

Question 10

In a batch of 100 items, the number of defective items follows a Poisson distribution with a mean of 5. What is the probability of finding exactly 3 defective items?

Accepted Answer

0.14

Answer

0.11

Answer

0.22

Answer

0.30

Question 11

The expected value of a binomial experiment with n trials and probability of success p is:

Accepted Answer

np

Answer

n(1-p)

Answer

np(1-p)

Answer

n^2p

Question 12

What is the probability mass function of a Poisson distribution with parameter λ?

Accepted Answer

(e^-λ)λ^x / x!

Answer

(1/e^λ)λ^x / x!

Answer

(λ/e^λ)x^λ

Answer

(e^λ/λ)x^λ

Question 13

Which of the following is NOT a characteristic of a binomial distribution?

Accepted Answer

Unbounded range

Answer

Discrete values

Answer

Fixed number of trials

Answer

Constant probability of success on each trial

Question 14

Which of the following statements about probability mass functions (PMFs) is true?

Accepted Answer

PMFs must sum to 1 over all possible outcomes.

Answer

PMFs can take negative values.

Answer

PMFs are always decreasing.

Answer

PMFs are only defined for continuous random variables.

Question 15

What is the probability mass function of the hypergeometric distribution?

Accepted Answer

P(X = x) = (C(x, K) * C(N - x, M - K)) / C(N, M)

Answer

P(X = x) = (x * K / N * M)

Answer

P(X = x) = (e^(-lambda) * lambda^x) / x!

Answer

P(X = x) = (M / N) ^ x * ((N - M) / (N - x))^ (K - x)

Question 16

What is the expected value of a Poisson distribution with a mean of 4?

Accepted Answer

4

Answer

2

Answer

8

Answer

16

Question 17

Which discrete probability distribution is used to model the number of successes in a sequence of independent trials?

Accepted Answer

Binomial distribution

Answer

Poisson distribution

Answer

Normal distribution

Answer

Hypergeometric distribution

Question 18

Which distribution is used to model the number of occurrences of an event in a fixed interval of time or space?

Accepted Answer

Poisson distribution

Answer

Binomial distribution

Answer

Negative binomial distribution

Answer

Geometric distribution

Question 19

The expected value of a Poisson distribution with parameter λ is:

Accepted Answer

λ

Answer

λ^2

Answer

1/λ

Answer

λ/2

Question 20

Which of the following is a discrete probability distribution that describes the probability of drawing k distinct objects from a population of N objects, without replacement?

Accepted Answer

Hypergeometric distribution

Answer

Binomial distribution

Answer

Geometric distribution

Answer

Negative binomial distribution

Question 21

Which of the following is a characteristic of discrete probability distributions?

Accepted Answer

They can only take a finite or countably infinite number of values.

Answer

Their values are continuous.

Answer

They can only be used for continuous data.

Answer

They have an infinite number of values.

Question 22

Which of the following is used to model the number of successes in a sequence of independent experiments with a constant probability of success?

Accepted Answer

Binomial distribution

Answer

Normal distribution is a continuous probability distribution.

Answer

Poisson distribution models the number of events that occur in a fixed interval of time or space.

Question 23

Which of the following criteria must be met for a distribution to be considered a discrete probability distribution?

Accepted Answer

It assigns probabilities to a countable set of distinct values.

Answer

Its mean and variance are finite.

Answer

Its values are all positive.

Question 24

A survey asks respondents to indicate their favorite color from a list of 6 options. What type of probability distribution is appropriate for modeling the number of respondents who choose each color?

Accepted Answer

Multinomial distribution

Answer

Poisson distribution models the number of events that occur in a fixed interval of time or space.

Answer

Uniform distribution assigns equal probabilities to all outcomes.

Answer

Binomial distribution is used to model the number of successes in a sequence of independent experiments with a constant probability of success.

Question 25

Which of the following is an example of a real-world application of the binomial distribution?

Accepted Answer

Predicting the number of defective parts in a manufacturing process.

Answer

Estimating the average height of a population. This is typically modeled using normal distribution.

Answer

Modeling the time until the next earthquake. This is typically modeled using exponential distribution.

Question 26

The binomial distribution is a discrete probability distribution used to model what phenomenon?

Accepted Answer

The number of successes in a series of independent experiments with a constant probability of success

Answer

The probability of drawing a specific number of successes from a population without replacement

Answer

The number of events occurring within a specified time interval or area

Answer

The number of successes in a population with a known proportion of successes

Question 27

In the context of a hypergeometric distribution, what factors determine the probability of drawing a specific number of successes?

Accepted Answer

The number of successes and failures in the population, as well as the sample size

Answer

Only the probability of success

Answer

Only the sample size

Question 28

Which assumption is fundamental to the binomial distribution?

Accepted Answer

The probability of success remains constant across all trials

Answer

The events are mutually exclusive

Answer

The number of trials is very large

Question 29

In a hypergeometric distribution, as the population size increases, the probability of drawing a specific number of successes approaches the probability of drawing that number of successes from what distribution?

Accepted Answer

Binomial distribution

Answer

Poisson distribution

Answer

Uniform distribution

Answer

Normal distribution

Question 30

The expected value (mean) of a binomial distribution with parameters n (number of trials) and p (probability of success) is:

Accepted Answer

n * p

Answer

n + p

Answer

n - p

Answer

n / p

Question 31

A company manufactures widgets with a 5% defect rate. In a batch of 50 widgets, what is the expected number of defective widgets?

Accepted Answer

2.5

Answer

0.25

Answer

10

Answer

5