Master Probability Distributions: Expected Value and Variance

Question 1

A discrete random variable X has a probability mass function defined as P(X = x) = k * x for x = 1, 2, 3, 4. Determine the value of the constant k.

Accepted Answer

1/10

Answer

1/5

Answer

1/4

Answer

1/2

Question 2

Which of the following is the formula for expected value of a random variable X with probability distribution P(X)?

Accepted Answer

E(X) = ΣxP(X)

Answer

E(X) = Σ(X - μ)²P(X)

Answer

E(X) = Σ(X + μ)²P(X)

Answer

E(X) = Σ(X - σ)²P(X)

Question 3

What is the expected value of a fair coin toss, where heads is assigned the value 1 and tails is assigned the value 0?

Accepted Answer

0.5

Answer

0

Answer

1

Answer

2

Question 4

Which of the following is the formula for variance of a random variable X with probability distribution P(X) and expected value μ?

Accepted Answer

Var(X) = E[(X - μ)²]

Answer

Var(X) = E[(X + μ)²]

Answer

Var(X) = Σ(X - μ)²P(X)

Answer

Var(X) = Σ(X + μ)²P(X)

Question 5

How are expected value and variance related?

Accepted Answer

Expected value is a measure of central tendency, while variance measures the spread.

Answer

Variance is a measure of central tendency, while expected value measures the spread.

Answer

Both expected value and variance measure central tendency.

Answer

Both expected value and variance measure the spread.

Question 6

Which of the following is the best measure of the typical value of a random variable?

Accepted Answer

Expected value

Answer

Mode

Answer

Median

Answer

Standard deviation

Question 7

When rolling a fair six-sided die, what is the variance of the random variable representing the number rolled?

Accepted Answer

2.92

Answer

1

Answer

5

Answer

18

Question 8

Given independent random variables X and Y with expected values of 5 and 10, respectively, what is the expected value of their sum Z = X + Y?

Accepted Answer

15

Answer

5

Answer

10

Answer

25

Question 9

From a population with a mean of 50 and a variance of 100, a sample of size 25 is drawn. What is the expected value of the sample mean?

Accepted Answer

50

Answer

25

Answer

75

Answer

100

Question 10

The variance of a binomial distribution is given by:

Accepted Answer

np(1-p)

Answer

np

Answer

np(1+p)

Answer

np^2

Question 11

For a random variable with a variance of 0.25, what is its standard deviation?

Accepted Answer

0.125

Answer

0.25

Answer

0.5

Answer

1

Question 12

Which of the following is a measure of central tendency?

Accepted Answer

Expected value

Answer

Variance

Answer

Standard deviation

Answer

Probability

Question 13

How is the expected value of a discrete random variable calculated?

Accepted Answer

Sum of all possible values multiplied by their probabilities

Answer

Average value of the random variable

Answer

Mode of the probability distribution

Answer

Probability of the most likely outcome

Question 14

Which of the following is the formula for calculating the expected value of a discrete random variable X?

Accepted Answer

E(X) = ∑x·p(x)

Answer

E(X) = ∫x·p(x)dx

Answer

E(X) = √x·p(x)dx

Answer

E(X) = 1/X

Question 15

A fair coin is flipped three times. What is the expected value of the number of heads obtained?

Accepted Answer

1.5

Answer

0.25

Answer

0.5

Answer

3

Question 16

The probability density function of a continuous random variable X is given by f(x) = (1/2)e^(-x/2), x > 0. What is the variance of X?

Accepted Answer

2

Answer

1

Answer

4

Answer

8

Question 17

A lottery has a jackpot of $1 million. The probability of winning the jackpot is 1 in 10,000,000. What is the expected value of buying one lottery ticket?

Accepted Answer

$0.10

Answer

$1.00

Answer

$10.00

Answer

$100.00

Question 18

The variance of a fair six-sided die roll is:

Accepted Answer

2/6

Answer

1/6

Answer

3/6

Answer

4/6

Question 19

Which of the following statements about the relationship between expected value and variance is correct?

Accepted Answer

Expected value can be negative, but variance cannot.

Answer

They are always equal.

Answer

Variance can be negative, but expected value cannot.

Answer

There is no relationship between them.

Question 20

Given a random variable X with expected value μ and variance σ^2, determine its standard deviation.

Accepted Answer

√(σ^2)

Answer

σ^2 - μ

Answer

μ - σ^2

Answer

√(μ^2)

Question 21

Define expected value as the:

Accepted Answer

Average outcome of a random experiment

Answer

Probability of a specific outcome in a random experiment

Answer

Difference between the expected value and variance

Answer

Sum of probabilities multiplied by random variable values

Question 22

A random variable X takes on values 1, 2, and 3 with probabilities 0.2, 0.5, and 0.3, respectively. Calculate the expected value of X:

Accepted Answer

2.0

Answer

1.7

Answer

2.2

Answer

2.5

Question 23

Which of the following best describes the variance of a random variable?

Accepted Answer

A measure of how spread out the probability distribution is

Answer

The average value of the random variable

Answer

The range of possible values of the random variable

Answer

The probability of obtaining any outcome

Question 24

For a continuous random variable with probability density function f(x), the expected value is given by:

Accepted Answer

∫xf(x)dx

Answer

∫x²f(x)dx

Answer

∫f(x)dx

Answer

∫xf²(x)dx

Question 25

Given a random variable X with expected value 5 and variance 4, what is its standard deviation?

Accepted Answer

2

Answer

1

Answer

4

Answer

8

Question 26

For independent random variables X and Y, the expected value of their sum X + Y is:

Accepted Answer

E(X) + E(Y)

Answer

E(X) - E(Y)

Answer

E(X) * E(Y)

Answer

E(X) / E(Y)

Question 27

The formula for calculating variance is:

Accepted Answer

Var(X) = E(X^2) - E(X)^2

Answer

Var(X) = E(X) / E(X^2)

Answer

Var(X) = ∑((x - E(X))^2 * P(x))

Answer

Var(X) = ∑(P(x) / (x - E(X))^2)

Question 28

What is the formula for calculating the expected value of a continuous random variable with probability density function f(x)?

Accepted Answer

∫xf(x)dx

Answer

∫f(x)/xdx

Answer

∫x^2f(x)dx

Answer

∫1/f(x)dx

Question 29

Consider two datasets with the same mean. Dataset A has a smaller variance than Dataset B. Which dataset has less spread in its data? Explain your reasoning.

Accepted Answer

Dataset A has less spread because its variance is smaller.

Answer

Dataset B has less spread because its variance is larger.

Answer

The spread of data in both datasets is the same because they have the same mean.

Answer

More information is needed to determine the spread of data in each dataset.

Question 30

In a casino game, you have a 40% chance of winning $5 and a 60% chance of losing $2. What is the expected value of this game?

Accepted Answer

$0.80

Answer

$1.00

Answer

$1.20

Question 31

You play a game where you have a 20% chance of winning $100 and an 80% chance of losing $20. What is the expected value of the game?

Accepted Answer

-$4

Answer

$20

Answer

-$20

Answer

$4

Question 32

The variance of a random variable X is defined as:

Accepted Answer

E[(X - μ)²]

Answer

E[X] + μ

Answer

E[X] - μ

Answer

μ

Question 33

Which of the following is NOT a property of variance?

Accepted Answer

It can be negative

Answer

It measures the spread of a distribution

Answer

It is always greater than or equal to zero

Question 34

In a binomial distribution with n = 10 and p = 0.5, what is the expected value?

Accepted Answer

5

Answer

0.25

Answer

0.5

Answer

10

Question 35

The expected value and variance of a continuous random variable X with probability density function f(x) are given by:

Accepted Answer

E[X] = ∫xf(x)dx and Var[X] = ∫(x - μ)²f(x)dx

Answer

E[X] = μ and Var[X] = σ²

Answer

E[X] = ∫f(x)dx and Var[X] = ∫x²f(x)dx

Answer

E[X] = ∫(x - μ)²f(x)dx and Var[X] = ∫xf(x)dx

Question 36

The variance of a dataset measures its:

Accepted Answer

Dispersion

Answer

Kurtosis

Answer

Skewness

Answer

Central tendency

Question 37

A normal distribution has a variance of 4. What is its standard deviation?

Accepted Answer

2

Answer

3

Answer

1

Answer

4

Question 38

Which of the following is NOT a characteristic of variance?

Accepted Answer

It can be negative.

Answer

It is always non-negative.

Answer

It describes the spread of a distribution.

Answer

It is measured in squared units.

Question 39

A survey of 50 students has a sample variance of 25 for their weights. Construct a 95% confidence interval for the true population variance.

Accepted Answer

(12.5, 45.5)

Answer

(10.0, 40.0)

Answer

(15.0, 50.0)

Answer

(20.0, 60.0)

Question 40

A fair six-sided die is rolled. What is the expected value of the number rolled?

Accepted Answer

3.5

Answer

3

Answer

4

Question 41

A company produces light bulbs with a mean lifespan of 1000 hours and a standard deviation of 50 hours. What is the variance of the lifespan?

Accepted Answer

2500

Answer

50

Answer

100

Answer

5000

Question 42

Which of the following statements is NOT true about the variance of a random variable?

Accepted Answer

It can be negative.

Answer

It measures the spread of the distribution.

Answer

It is calculated as the average squared deviation from the mean.

Answer

It is always non-negative.

Question 43

If two random variables X and Y are independent, what is the variance of their sum, Var(X + Y)?

Accepted Answer

Var(X) + Var(Y)

Answer

Var(X) - Var(Y)

Answer

(Var(X) + Var(Y)) / 2

Answer

Var(X) * Var(Y)

Question 44

The expected value of a constant 'c' times a random variable X is equal to:

Accepted Answer

c times the expected value of X.

Answer

The expected value of X divided by c.

Answer

c plus the expected value of X.

Question 45

Which of the following scenarios would benefit from understanding expected value?

Accepted Answer

An investor deciding between different stock portfolios.

Answer

A musician tuning an instrument.

Answer

A chef measuring ingredients for a cake.

Question 46

A fair coin is flipped three times. Calculate the expected value of the number of heads.

Accepted Answer

1.5

Answer

1

Answer

2

Answer

3

Question 47

A game involves rolling a six-sided die. You win $5 if you roll a 6, lose $1 if you roll a 1 or 2, and win nothing otherwise. Determine the expected value of this game.

Accepted Answer

$0.50

Answer

$1.00

Answer

$-0.50

Answer

$0.00

Question 48

Given the probability distribution of a random variable X:

| X | P(X) |
|---|---| 
| 0 | 0.2 |
| 1 | 0.3 |
| 2 | 0.4 |
| 3 | 0.1 |

Calculate the variance of X.

Accepted Answer

1.06

Answer

0.54

Answer

1.50

Answer

0.84

Question 49

A random variable X has a mean of 10 and a standard deviation of 2. Calculate the variance of X.

Accepted Answer

4

Answer

2

Answer

16

Answer

10

Question 50

Which of the following statements about variance is accurate?

Accepted Answer

Variance is always non-negative.

Answer

Variance is always positive.

Answer

Variance is a measure of central tendency.

Answer

Variance is a measure of the shape of a distribution.

Question 51

Two random variables, X and Y, have identical expected values. Which of the following statements about their variances is true?

Accepted Answer

Their variances can be different.

Answer

Their variances must be the same.

Answer

The variance of Y must be greater than the variance of X.

Answer

The variance of X must be greater than the variance of Y.

Question 52

A company manufactures light bulbs with a mean lifespan of 1000 hours and a standard deviation of 50 hours. Determine the variance of the lifespan of these light bulbs.

Accepted Answer

2500

Answer

5000

Answer

1000

Answer

50

Question 53

A random variable X has a probability distribution with a mean of 5 and a variance of 2. Calculate the expected value of X^2.

Accepted Answer

27

Answer

10

Answer

25

Question 54

A student takes a multiple-choice test with 10 questions, each having 4 options with only one correct answer. If the student randomly guesses on all questions, calculate the expected number of correct answers.

Accepted Answer

2.5

Answer

10

Answer

5

Answer

1

Question 55

Which scenario is most likely to have the highest variance?

Accepted Answer

The height of randomly selected adult males in a city.

Answer

The number of heads when flipping a coin 10 times.

Answer

The temperature in a room over a 24-hour period.