Random Variables: Probability Distributions and Applications

Question 1

Which of the following probability distributions is used to model the time between arrivals in a queuing system?

Accepted Answer

Exponential distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Normal distribution

Question 2

Which of the following sampling methods involves randomly selecting a fixed proportion of the population?

Accepted Answer

Simple random sampling

Answer

Systematic sampling

Answer

Stratified sampling

Answer

Cluster sampling

Question 3

Which of the following measures of central tendency is least affected by outliers?

Accepted Answer

Median

Answer

Mean

Answer

Mode

Answer

Standard deviation

Question 4

What is the probability of occurrence of a particular value of a random variable X, denoted as P(X = x)?

Accepted Answer

P(X = x)

Answer

f(X = x)

Answer

μ(X = x)

Answer

σ(X = x)

Question 5

What is the variance of a binomial random variable with parameters n and p?

Accepted Answer

np(1-p)

Answer

np(1+p)

Answer

np^2

Answer

(np)^2

Question 6

What is the probability density function of a continuous random variable X with uniform distribution on the interval (0, 1)?

Accepted Answer

f(x) = 1

Answer

f(x) = x

Answer

f(x) = 1/x

Answer

f(x) = x^2

Question 7

What is the expected value of a random variable X following a Poisson distribution with parameter λ?

Accepted Answer

λ

Answer

λ^2

Answer

e^λ

Answer

1/λ

Question 8

Which statistical hypothesis test is used to compare the means of two independent populations?

Accepted Answer

t-test

Answer

χ^2 test

Answer

ANOVA

Answer

Kruskal-Wallis test

Question 9

In a manufacturing process, the lifetime of items follows an exponential distribution with a mean of 500 hours. What is the probability that an item will fail within the first 250 hours of operation?

Accepted Answer

0.25

Answer

0.50

Answer

0.75

Answer

1.00

Question 10

Which characteristic does NOT apply to a random variable?

Accepted Answer

Has a fixed value

Answer

Assumes numerical values

Answer

Represents an outcome of an experiment

Answer

Can be either discrete or continuous

Question 11

Identify a type of continuous probability distribution:

Accepted Answer

Normal

Answer

Binomial

Answer

Poisson

Answer

Hypergeometric

Question 12

Which of the following measures describes the spread or dispersion of a random variable?

Accepted Answer

Variance

Answer

Mean

Answer

Median

Answer

Range

Question 13

When a fair coin is flipped twice, what is the probability of obtaining heads on both flips?

Accepted Answer

1/4

Answer

1/2

Answer

1

Answer

2

Question 14

A random variable X follows a normal distribution with a mean of 50 and a standard deviation of 10. Calculate the probability that X falls between 40 and 60.

Accepted Answer

0.6826

Answer

0.3413

Answer

0.9544

Answer

1

Question 15

Which of the following is not a type of random variable?

Accepted Answer

Constant

Answer

Continuous

Answer

Categorical

Answer

Discrete

Question 16

Which of the following is a measure of central tendency?

Accepted Answer

Mean

Answer

Variance

Answer

Kurtosis

Answer

Standard deviation

Question 17

Let X be a random variable with a uniform distribution on the interval [0, 1]. What is the expected value of X?

Accepted Answer

1/2

Answer

0

Answer

1

Question 18

What is the probability mass function (PMF) for a Bernoulli random variable with parameter *p*?

Accepted Answer

P(X=1) = *p*, P(X=0) = 1-*p*

Answer

P(X=1) = 1-*p*, P(X=0) = *p*

Answer

P(X=1) = 1, P(X=0) = 0

Answer

P(X=1) = *p*, P(X=0) = *p*

Question 19

A random variable *X* follows a binomial distribution with parameters *n* = 5 and *p* = 0.4.  What is the probability of obtaining exactly 3 successes?

Accepted Answer

0.256

Answer

0.32

Answer

0.512

Answer

0.4

Question 20

What is the expected value (mean) of a Poisson random variable with parameter λ = 3?

Accepted Answer

3

Answer

2

Answer

4

Answer

5

Question 21

What is the variance of a uniform random variable defined over the interval [a, b]?

Accepted Answer

(b-a)²/12

Answer

(b-a)²/4

Answer

(b-a)²

Answer

(b-a)²/2

Question 22

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

Accepted Answer

Height of students in a class

Answer

Number of heads in 10 coin flips

Answer

Grade on a multiple-choice test

Answer

Roll of a fair dice

Question 23

The probability density function (PDF) of a normal random variable is symmetrical about which point?

Accepted Answer

Its mean

Answer

Its variance

Answer

Its median

Answer

Its mode

Question 24

Which of the following is a measure of central tendency for a dataset?

Accepted Answer

Mean

Answer

Variance

Answer

Skewness

Answer

Standard deviation

Question 25

Which of the following is NOT a property of the joint probability distribution of two random variables *X* and *Y*?

Accepted Answer

P(X=x) + P(Y=y) = 1

Answer

P(X=x|Y=y) = P(Y=y|X=x)

Answer

∑x∑y P(X=x, Y=y) = 1

Answer

P(X=x, Y=y) >= 0

Question 26

In a conditional probability statement P(A|B), what is B referred to as?

Accepted Answer

Condition

Answer

Event

Answer

Outcome

Answer

Hypothesis