Master Random Variables in Engineering Mathematics: Interactive Quizzes

Question 1

The probability density function (PDF) of a random variable X is given by f(x) = λe^(-λx), for x ≥ 0. What is the expected value of X?

Accepted Answer

1/λ

Answer

1

Answer

λ

Answer

λ^2

Question 2

Which of the following is true about the variance of a random variable?

Accepted Answer

It measures the spread or dispersion of the random variable around its mean.

Answer

It can be negative.

Answer

It is always less than or equal to the expected value.

Answer

It is a dimensionless quantity.

Question 3

Which of the following is a moment generating function of a random variable?

Accepted Answer

E(e^tX)

Answer

E(X^n)

Answer

E(X-μ)^n

Answer

E(X^2+μ^2)

Question 4

In probability theory, which term represents the average value of a random variable?

Accepted Answer

Expected value

Answer

Variance

Answer

Probability density function

Answer

Cumulative distribution function

Question 5

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

Accepted Answer

It can be negative for some distributions.

Answer

It measures the spread of the distribution.

Answer

It is always non-negative.

Answer

It is the square of the standard deviation.

Question 6

Two random variables X and Y are independent if:

Accepted Answer

P(X = x, Y = y) = P(X = x) * P(Y = y) for all x and y

Answer

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

Answer

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

Answer

E(XY) = E(X) * E(Y)

Question 7

The covariance between two random variables X and Y measures:

Accepted Answer

The strength and direction of their linear relationship.

Answer

The spread of their distribution.

Answer

The probability of their intersection.

Answer

The difference between their means.

Question 8

Which of the following distributions is used to model the time between failures in a system?

Accepted Answer

Exponential distribution

Answer

Normal distribution

Answer

Poisson distribution

Answer

Binomial distribution

Question 9

According to the central limit theorem, for a large enough sample size, the distribution of sample means from any population will be approximately:

Accepted Answer

Normal

Answer

Uniform

Answer

Exponential

Question 10

Which of the following is **not** a feature of a random variable?

Accepted Answer

Always follows a continuous probability distribution

Answer

Assumes numerical values

Answer

Represents the outcome of an uncertain event

Answer

Can have multiple possible values

Question 11

What is the skewness of a standard normal distribution?

Accepted Answer

0

Answer

1

Answer

-1

Answer

2

Question 12

Which of these options represents a joint probability distribution?

Accepted Answer

P(X = x, Y = y)

Answer

P(X = x)

Answer

P(X < x)

Answer

P(X > x)

Question 13

What is the correlation coefficient between two independent random variables?

Accepted Answer

0

Answer

1

Answer

-1

Answer

2

Question 14

Which of the following is the variance of a binomial random variable X with parameters n = 12 and p = 0.3?

Accepted Answer

2.52

Answer

4.52

Answer

1.52

Answer

3.52

Question 15

A random variable X has PDF f(x) = x for 0 <= x <= 1. What is the probability that X takes a value greater than 0.5?

Accepted Answer

0.25

Answer

0.5

Answer

1

Answer

0.75

Question 16

A random variable X has PDF f(x) = (1/√(2π))exp[-(x^2)/2]. What type of distribution does X follow?

Accepted Answer

Standard normal distribution

Answer

Exponential distribution

Answer

Poisson distribution

Answer

Binomial distribution

Question 17

Which of the following statements about the moments of a random variable X is correct?

Accepted Answer

The k-th moment of X is defined as E[(X - μ)^k], where μ is the mean of X

Answer

The k-th moment of X is defined as (E[X])^k

Answer

The k-th moment of X is defined as E[(X + μ)^k]

Answer

The k-th moment of X is defined as (Var[X])^k

Question 18

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

Accepted Answer

Deterministic variable

Answer

Continuous random variable

Answer

Discrete random variable

Question 19

Two random variables X and Y have the joint PMF P(X = x, Y = y) = (1/9) for x = 1, 2, 3 and y = 1, 2, 3. What is the variance of the sum of X and Y?

Accepted Answer

2

Answer

3

Answer

1/3

Answer

1

Question 20

Two independent random variables X and Y have standard normal distributions. The probability that X > 0 and Y < 0 is:

Accepted Answer

0.25

Answer

0.5

Answer

0.75

Answer

0.125

Question 21

A random variable X has a uniform distribution between a and b. What is the expected value of X?

Accepted Answer

(a + b)/2

Answer

a

Answer

0

Answer

b

Question 22

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

Accepted Answer

Σ[x * P(X = x)]

Answer

x - P(X = x)

Answer

P(X = x)/x

Answer

∫[x * P(X = x)]dx

Question 23

What is the variance of a continuous random variable X with probability density function f(x)?

Accepted Answer

∫[(x - μ)² * f(x)]dx

Answer

Σ[(x - μ)² * P(X = x)]

Answer

(μ - x)²

Answer

∫[(x + μ)² * f(x)]dx

Question 24

What is the mean of a uniform distribution over the interval [a, b]?

Accepted Answer

(a + b)/2

Answer

(a - b)/2

Answer

(a * b)/2

Question 25

Which of the following is a characteristic of a normal distribution?

Accepted Answer

Symmetric, bell-shaped curve

Answer

Asymmetrical curve

Answer

Triangular curve

Answer

Rectangular curve

Question 26

Which of the following is a measure of the spread of a distribution?

Accepted Answer

Variance

Answer

Mode

Answer

Median

Answer

Mean

Question 27

What is the relationship between the variance and standard deviation of a random variable?

Accepted Answer

Standard deviation is the square root of variance

Answer

Variance is the square root of standard deviation

Answer

Variance is the square of standard deviation

Answer

Standard deviation is the variance