Master Random Variables: Probability and Statistics for Mathematics Majors

Question 1

What is the definition of a probability mass function (PMF) for a discrete random variable?

Accepted Answer

A function that assigns a probability to each possible value of the random variable.

Answer

A function that assigns the total probability of all possible values of the random variable to 1.

Answer

A function that calculates the expected value of the random variable.

Answer

A function that calculates the variance of the random variable.

Question 2

Which of the following is NOT an example of a probability distribution?

Accepted Answer

Standard deviation

Answer

Normal distribution

Answer

Binomial distribution

Answer

Exponential distribution

Question 3

What does the expected value of a random variable represent?

Accepted Answer

The average value of the random variable.

Answer

The most frequent value of the random variable.

Answer

The probability of the random variable taking on a particular value.

Answer

The spread of the random variable.

Question 4

What is the formula for calculating the variance of a random variable X?

Accepted Answer

E[X^2] - (E[X])^2

Answer

E[(X - μ)^2]

Answer

σ^2 * μ

Answer

P(X = x)^2

Question 5

What is the relationship between a random variable and a random experiment?

Accepted Answer

A random variable is a function of a random experiment.

Answer

A random experiment is a function of a random variable.

Answer

They are the same thing.

Answer

They are unrelated.

Question 6

What is the probability mass function (PMF) of a discrete random variable?

Accepted Answer

Assigns a probability to each possible value of the random variable.

Answer

Assigns a probability to each interval of values of the random variable.

Answer

Represents the total probability of all possible outcomes.

Answer

Equals the expected value of the random variable.

Question 7

What defines the probability density function (PDF) of a continuous random variable?

Accepted Answer

Assigns a probability to each interval of values of the random variable.

Answer

Assigns a probability to each possible value of the random variable.

Answer

Represents the total probability of all possible outcomes.

Answer

Equals the expected value of the random variable.

Question 8

What is the expected value of a random variable?

Accepted Answer

Average value of the random variable.

Answer

Most probable value of the random variable.

Answer

Probability of the random variable taking on a particular value.

Answer

Standard deviation of the random variable.

Question 9

What does the variance of a random variable measure?

Accepted Answer

Spread or dispersion of the random variable.

Answer

Probability of the random variable taking on a particular value.

Answer

Expected value of the random variable.

Answer

Range of the random variable.

Question 10

What is the relationship between variance and standard deviation?

Accepted Answer

Standard deviation is the square root of variance.

Answer

Standard deviation is equal to variance.

Answer

Variance is the square root of standard deviation.

Answer

Standard deviation is the absolute value of variance.

Question 11

Which distribution is suitable for modeling the number of successes in a fixed number of independent trials with a constant probability of success?

Accepted Answer

Binomial distribution

Answer

Poisson distribution

Answer

Normal distribution

Answer

Exponential distribution

Question 12

Which distribution is used to model the number of events occurring in a fixed interval of time or space when the occurrence of each event is independent of the others?

Accepted Answer

Poisson distribution

Answer

Binomial distribution

Answer

Normal distribution

Answer

Uniform distribution

Question 13

Which distribution is often used to model continuous random variables, such as heights or weights, that follow a bell-shaped curve?

Accepted Answer

Normal distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Chi-square distribution

Question 14

Which of the following is a continuous random variable?

Accepted Answer

B. Height of students in a class

Answer

A. Number of heads in 10 coin flips

Answer

C. Number of siblings a person has

Question 15

What is the probability mass function of a Bernoulli random variable with parameter p?

Accepted Answer

f(x) = p^x * (1-p)^(1-x)

Answer

f(x) = x * p * (1-p)^(x-1)

Answer

f(x) = p * (1-p)^x

Answer

f(x) = (1-p)^x

Question 16

Which of the following is the probability density function of a continuous random variable with probability density function f(x)?

Accepted Answer

P(a ≤ X ≤ b) = ∫[a,b] f(x) dx

Answer

P(X = x) = f(x)

Answer

E(X) = ∫[a,b] x * f(x) dx

Answer

Var(X) = ∫[a,b] (x - E(X))^2 * f(x) dx

Question 17

What is the expected value of a random variable X with probability mass function f(x)?

Accepted Answer

E(X) = ∑[x] x * f(x)

Answer

E(X) = ∫[a,b] x * f(x) dx

Answer

E(X) = Var(X)

Answer

E(X) = P(X = x)

Question 18

What is the mean of a random variable X with probability density function f(x) = 1/2 for 0 ≤ x ≤ 2?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 19

Consider a random variable X with probability density function f(x). Which property accurately describes the cumulative distribution function (CDF) of X?

Accepted Answer

The CDF of X is non-decreasing.

Accepted Answer

The CDF of X ranges from 0 to 1.

Answer

The CDF of X is continuous at every point.

Answer

The CDF of X is symmetric around its mean.

Question 20

How is the cumulative distribution function (CDF) of a continuous random variable mathematically related to its probability density function (PDF)?

Accepted Answer

The CDF is the integral of the PDF over the entire real line.

Answer

The CDF is the derivative of the PDF with respect to the random variable.

Answer

The CDF is the inverse function of the PDF.

Answer

The CDF and PDF are unrelated concepts in probability theory.

Question 21

Which characteristic is NOT associated with a random variable?

Accepted Answer

Deterministic

Answer

Dependent on the experiment's outcome

Answer

Probability distribution

Answer

Numerical outcome

Question 22

Discrete random variables are characterized by which type of probability?

Accepted Answer

Point probabilities

Answer

Joint probabilities

Answer

Continuous probabilities

Question 23

A continuous random variable is associated with which function?

Accepted Answer

Probability density function

Answer

Probability mass function

Answer

Cumulative distribution function

Answer

Joint probability distribution

Question 24

Which of the following is NOT a measure of dispersion for a random variable?

Accepted Answer

Median

Answer

Interquartile range

Answer

Variance

Answer

Standard deviation

Question 25

State the central limit theorem.

Accepted Answer

As the sample size increases, the distribution of sample means approaches a normal distribution.

Answer

The probability of an event is directly proportional to its relative frequency.

Answer

The expected value and median of a random variable are always equal.

Question 26

Which property is characteristic of independent random variables?

Accepted Answer

The probability of their joint occurrence is equal to the product of their individual probabilities.

Answer

Their means are always equal.

Answer

Their variances are necessarily identical.

Question 27

For a random experiment of rolling a fair six-sided die once, let X represent the number on the top face. Which of the following is the probability mass function of X?

Accepted Answer

P(X = x) = 1/6, for x = 1, 2, 3, 4, 5, 6

Answer

P(X = x) = 1/3, for x = 1, 2, 3

Answer

P(X = x) = x/6, for x = 1, 2, 3, 4, 5, 6

Question 28

Let X be a continuous random variable with probability density function f(x) = 2x, for 0 ≤ x ≤ 1. What is the expected value of X?

Accepted Answer

E(X) = 2/3

Answer

E(X) = 1/3

Answer

E(X) = 1

Answer

E(X) = 1/2

Question 29

Let X and Y be random variables with E(X) = 0, E(Y) = 1, and Cov(X, Y) = 0.5. What is the correlation coefficient between X and Y?

Accepted Answer

0.5

Answer

1

Answer

-0.5

Answer

0

Question 30

Which of the following is NOT a characteristic of a random variable?

Accepted Answer

A random variable is a constant value.

Answer

A random variable can be discrete or continuous.

Answer

A random variable describes the probability distribution of an experiment.

Answer

A random variable assigns a numerical value to each outcome.

Question 31

A random variable X follows a normal distribution with mean μ and standard deviation σ. What is the probability that X lies within one standard deviation of the mean?

Accepted Answer

0.6827

Answer

0.8413

Answer

0.9545

Answer

0.5000

Question 32

Which of the following is a necessary condition for two random variables to be independent?

Accepted Answer

Their joint probability distribution is equal to the product of their individual probability distributions.

Answer

Their variances are equal.

Answer

Their expected values are equal.

Question 33

Which of the following is NOT a property of a random variable?

Accepted Answer

It must have an infinite range.

Answer

It must be a function that assigns a numerical value to each outcome.

Answer

It must have a range of possible values.

Answer

It must have a probability distribution.

Question 34

Which of the following is the probability mass function of the binomial distribution?

Accepted Answer

P(X = x) = (n!/(x!(n-x)!)) * p^x * (1-p)^(n-x)

Answer

P(X = x) = (n!/(x!(n-x)!)) * q^x * (1-q)^(n-x)

Answer

P(X = x) = (n!/(x!(n+x)!)) * p^x * (1-p)^(n-x)

Answer

P(X = x) = (n!/(x!(n+x)!)) * q^x * (1-q)^(n+x)

Question 35

Which of the following is NOT a method of generating random numbers?

Accepted Answer

Using a deterministic algorithm

Answer

Using a statistical table

Answer

Using a pseudorandom number generator

Answer

Using a physical random number generator

Question 36

What is the relationship between the mean and median of a distribution?

Accepted Answer

The mean may not be equal to the median.

Answer

The mean is always less than or equal to the median.

Answer

The mean and median are always equal.

Question 37

What is a probability mass function (PMF) for a discrete random variable?

Accepted Answer

A function that assigns a probability to each possible value of the random variable.

Answer

A function that assigns a probability to each range of values of the random variable.

Answer

A function that assigns a probability to each outcome of a random experiment.

Question 38

What is a probability density function (PDF) for a continuous random variable?

Accepted Answer

A function that assigns a probability to each range of values of the random variable.

Answer

A function that assigns a probability to each possible value of the random variable.

Answer

A function that assigns a probability to each outcome of a random experiment.

Question 39

Which of the following is a property of a probability density function?

Accepted Answer

The integral of the PDF over the entire range of possible values of the random variable is 1.

Answer

The PDF is always a positive function.

Answer

The probability of any one value of the random variable is always greater than 0.

Question 40

What is the expected value of a random variable?

Accepted Answer

The mean or average of the random variable.

Answer

The most likely value of the random variable.

Answer

The variance of the random variable.

Question 41

Which of the following is NOT a property of a discrete random variable?

Accepted Answer

Its probability mass function is continuous.

Answer

It can take on any value within a given range.

Answer

The sum of its probabilities is 1.

Answer

Its values are countable.

Question 42

A random variable X has the following probability mass function:
P(X = 0) = 0.4, P(X = 1) = 0.3, P(X = 2) = 0.2, P(X = 3) = 0.1.
What is the expected value of X?

Accepted Answer

1.2

Answer

1.6

Answer

0.8

Question 43

Which of the following is a valid probability density function for a continuous random variable?

Accepted Answer

f(x) = e^(-x) for x >= 0

Answer

f(x) = 1/x for x between 1 and 2

Answer

f(x) = x for x between 0 and 1

Question 44

A continuous random variable X has probability density function f(x) = 2x for 0 <= x <= 1.
What is the probability that X is less than 0.5?

Accepted Answer

0.25

Answer

1.0

Answer

0.75

Answer

0.5

Question 45

Which of the following transformations of a random variable preserves its distribution?

Accepted Answer

Linear transformation: Y = aX + b

Answer

Power transformation: Y = X^2

Answer

Exponential transformation: Y = e^X

Answer

Logarithmic transformation: Y = log(X)

Question 46

Let X and Y be two independent random variables with means μX and μY, respectively.
What is the mean of the random variable Z = X + Y?

Accepted Answer

μX + μY

Answer

μX - μY

Answer

μX / μY

Answer

μX * μY

Question 47

If a random variable *X* has a variance of 9, what is the variance of the random variable *Y* = 2*X* + 3?

Accepted Answer

36

Answer

9

Answer

81

Answer

18

Question 48

Which probability distribution is commonly used to model the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes?

Accepted Answer

Binomial distribution

Answer

Poisson distribution

Answer

Normal distribution

Answer

Uniform distribution

Question 49

A bag contains 5 red balls and 3 blue balls. If two balls are drawn randomly without replacement, what is the probability that both balls are red?

Accepted Answer

2/7

Answer

1/3

Answer

1/2

Answer

1/10

Question 50

Which of the following is NOT a characteristic of a random variable?

Accepted Answer

It is a real-valued function

Answer

It can be either discrete or continuous

Answer

It assigns a numerical value to each outcome

Answer

It is defined on the sample space of an experiment

Question 51

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

Accepted Answer

Number of heads in 10 coin tosses

Answer

Weight of a randomly selected car

Answer

Time until the next earthquake

Answer

Height of a randomly selected person

Question 52

The probability mass function (PMF) of a discrete random variable *X* gives the probability of each possible value. If *X* takes on the values 1, 2, and 3 with probabilities 0.2, 0.5, and 0.3, respectively, what is its PMF?

Accepted Answer

P(X = 1) = 0.2, P(X = 2) = 0.5, P(X = 3) = 0.3

Answer

P(X = 1) = 0.5, P(X = 2) = 0.2, P(X = 3) = 0.3

Answer

P(X = 1) = 0.3, P(X = 2) = 0.5, P(X = 3) = 0.2

Answer

P(X = 1) = 0.2, P(X = 2) = 0.3, P(X = 3) = 0.5

Question 53

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

Accepted Answer

Height of a randomly selected person

Answer

Number of heads in 10 coin tosses

Answer

Weight of a randomly selected car

Answer

Time until the next earthquake

Question 54

The probability density function (PDF) of a continuous random variable gives the probability of the random variable taking on a specific value. What is the PDF of the random variable *X*, which is uniformly distributed between 0 and 1?

Accepted Answer

f(x) = 1 for 0 ≤ x ≤ 1

Answer

f(x) = x for 0 ≤ x ≤ 1

Answer

f(x) = x^2 for 0 ≤ x ≤ 1

Answer

f(x) = 1/x for 0 ≤ x ≤ 1

Question 55

The expected value (mean) of a random variable is a measure of the central tendency of its distribution. What is the expected value of the random variable *X*, which takes on the values 1, 2, and 3 with probabilities 0.2, 0.5, and 0.3, respectively?

Accepted Answer

E(X) = 2

Answer

E(X) = 1

Answer

E(X) = 3

Answer

E(X) = 4

Question 56

The variance of a random variable measures the spread of its distribution. What is the variance of the random variable *X*, which has a PMF of P(X = 1) = 0.2, P(X = 2) = 0.5, and P(X = 3) = 0.3?

Accepted Answer

Var(X) = 0.8

Answer

Var(X) = 2

Answer

Var(X) = 3

Answer

Var(X) = 0.2

Question 57

The standard deviation of a random variable is a measure of the spread of the distribution. What is the standard deviation of the random variable X, which has a variance of 1?

Accepted Answer

SD(X) = 1

Answer

SD(X) = 2

Answer

SD(X) = 0.2

Answer

SD(X) = 3

Question 58

The cumulative distribution function (CDF) of a random variable gives the probability that the random variable takes on a value less than or equal to a specific value. What is the CDF of the random variable X, which takes on the values 1, 2, and 3 with probabilities 0.2, 0.5, and 0.3, respectively?

Accepted Answer

F(x) = 0 for x < 1, F(x) = 0.2 for 1 ≤ x < 2, F(x) = 0.7 for 2 ≤ x < 3, F(x) = 1 for x ≥ 3

Answer

F(x) = 0.2 for x < 1, F(x) = 0.7 for 1 ≤ x < 2, F(x) = 1 for 2 ≤ x < 3, F(x) = 0 for x ≥ 3

Answer

F(x) = 0 for x ≤ 1, F(x) = 0.2 for 1 < x ≤ 2, F(x) = 0.7 for 2 < x ≤ 3, F(x) = 1 for x > 3

Question 59

The median of a random variable is the value that divides the distribution into two equal parts. What is the median of the random variable X, which takes on the values 1, 2, and 3 with probabilities 0.2, 0.5, and 0.3, respectively?

Accepted Answer

Median = 2

Answer

Median = 3

Answer

Median = 4

Answer

Median = 1