Probability Theory Quizzes: Test Your Knowledge in Probability and Statistics

Question 1

Independence of two events A and B means that:

Accepted Answer

P(A ∩ B) = P(A)P(B)

Answer

P(A|B) = P(A)

Answer

P(A) = P(B)

Answer

P(A|B) = P(B)

Question 2

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

Accepted Answer

Has a normal distribution curve

Answer

Discrete

Answer

Takes values between 0 and n

Answer

Mean of np

Question 3

The central limit theorem states that the distribution of sample means will be:

Accepted Answer

Approximately normal

Answer

Binomial

Answer

Poisson

Answer

Uniform

Question 4

Which of the following is NOT an axiom of probability theory?

Accepted Answer

The probability of the union of any two events is the sum of their probabilities.

Answer

The probability of an event occurring is a non-negative number between 0 and 1.

Answer

The probability of the complement of an event is 1 minus the probability of the event.

Answer

The probability of an impossible event is 0.

Question 5

Which of the following is a discrete random variable?

Accepted Answer

The number of cars passing by a traffic light in an hour

Answer

The time until the next earthquake

Answer

The height of a randomly selected person

Answer

The weight of a newborn baby

Question 6

Which of the following is a fundamental property of a probability measure?

Accepted Answer

It sums up to 1.

Answer

It is always positive.

Answer

It sums up to zero.

Answer

It is continuous.

Question 7

Which of the following is a measure of dispersion?

Accepted Answer

Standard deviation

Answer

Mean

Answer

Median

Answer

Mode

Question 8

Which of the following is NOT a sampling method that ensures a random selection of participants?

Accepted Answer

Convenience sampling

Answer

Simple random sampling

Answer

Stratified sampling

Answer

Systematic sampling

Question 9

Which of the following probability distributions is most suitable for modeling the time between arrivals at a service desk?

Accepted Answer

Exponential distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Normal distribution

Question 10

Which distribution is symmetric around its mean?

Accepted Answer

Normal distribution

Answer

Uniform distribution

Answer

Exponential distribution

Answer

Binomial distribution

Question 11

Let X be a continuous random variable with probability density function f(x) = 1/βe^(-x/β), x > 0, where β > 0. What is the mean of X?

Accepted Answer

β

Answer

β/2

Answer

β/4

Answer

β/8

Question 12

Let X1, X2, ..., Xn be a random sample from a normal distribution with mean μ and variance σ². What is the distribution of the sample average?

Accepted Answer

Normal distribution

Answer

Student's t-distribution

Answer

Chi-squared distribution

Answer

Fisher's F-distribution

Question 13

Let X and Y be two discrete random variables such that P(X = x, Y = y) = (1/2)^x(1/4)^y, x = 0, 1, 2, ..., y = 0, 1, 2, .... What is the marginal probability mass function of X?

Accepted Answer

P(X = x) = (1/2)^x

Answer

P(X = x) = (1/2)^x(1/2)^y

Answer

P(X = x) = (1/4)^x(1/2)^y

Answer

P(X = x) = (1/4)^x

Question 14

Let X be a discrete random variable. What is the formula for its expected value?

Accepted Answer

∑[x * p(x)]

Answer

∫[x * f(x) dx]

Answer

Max[p(x)]

Answer

Mode[p(x)]

Question 15

Which of the following distributions is a continuous probability distribution?

Accepted Answer

Normal distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Hypergeometric distribution

Question 16

Let X and Y be two independent random variables. What is the probability that both X = 2 and Y = 3?

Accepted Answer

p(X=2) * p(Y=3)

Answer

p(X=2) + p(Y=3)

Answer

p(X=2) - p(Y=3)

Answer

0

Question 17

Let X be a normally distributed random variable with mean μ = 10 and standard deviation σ = 2. What is the probability that X is less than 6?

Accepted Answer

0.0228

Answer

0.05

Answer

0.1587

Answer

0.9772

Question 18

Which of the following is a key principle of Bayesian inference?

Accepted Answer

It updates probabilities based on new evidence.

Answer

It assumes a fixed prior probability distribution.

Answer

It is only applicable to discrete random variables.

Answer

It requires a large sample size to be accurate.

Question 19

Which of the following is NOT a type of probability sampling method?

Accepted Answer

Quota sampling

Answer

Simple random sampling

Answer

Stratified sampling

Answer

Cluster sampling

Question 20

Which of the following functions is a valid moment generating function (mgf)?

Accepted Answer

e^(x^2)

Answer

sin(x)

Answer

1/(1-x)

Answer

x^3

Question 21

Which of the following is an example of a continuous probability distribution?

Accepted Answer

Normal distribution

Answer

Poisson distribution

Answer

Binomial distribution

Answer

Hypergeometric distribution

Question 22

A random variable X has a probability mass function given by P(X = x) = (1/2)^x for x = 1, 2, 3, …. Which distribution does X follow?

Accepted Answer

Geometric

Answer

Binomial

Answer

Poisson

Answer

Exponential

Question 23

A fair coin is flipped twice. What is the probability of getting at least one tail?

Accepted Answer

3/4

Answer

1/2

Answer

1/4

Answer

1/8

Question 24

Which of the following is NOT a property of probability measures?

Accepted Answer

P(A ⋂ B) = P(A) - P(B)

Answer

P(E) >= 0 for all events E

Answer

P(S) = 1

Answer

P(A ∪ B) = P(A) + P(B)

Question 25

Which of the following is an example of a discrete random variable?

Accepted Answer

Number of accidents per day

Answer

Height of a person

Answer

Temperature of a room

Answer

Average rainfall per year