A 6-sided die is rolled once. What is the expected value of the outcome of the roll?

Choose one answer.

	a. 3
	b. 3.5
	c. 21
	d. 5.5

Question 2

A continuous random variable X has the following cumulative distribution function: F(x) = x/5 for x between 0 and 5, F(x) = 0 for x < 0, and F(x) = 1 for x > 5. What is the probability of 4 ≥ X ≥ 3?

Choose one answer.

	a. 0.25
	b. 0.40
	c. 0.15
	d. None of the above

Question 3

A continuous random variable X has the following cumulative distribution function: F(x) = x²/4 for x between 0 and 2, F(x) = 0 for x < 0, and F(x) = 1 for x > 2. What is the probability of X ≥ 1?

Choose one answer.

	a. 0.75
	b. 0.40
	c. 0.15
	d. None of the above

Question 4

A die is rigged so that it only rolls out even numbers. What is the sample space of the experiment in which the die is rolled once?

Choose one answer.

	a. {1, 2, 3, 4, 5, 6}
	b. {2, 4, 6}
	c. {1, 2, 3}
	d. {2, 3, 4, 5}

Question 5

Brenda flipped a coin four times. What is the probability that she had at least one head out of four tosses?

Choose one answer.

	a. 0.25
	b. 0.9375
	c. 0.75
	d. 1

Question 6

Calculate the probability density function at x = 1 for an exponential distribution with λ = 1.

Choose one answer.

	a. 0.1825
	b. 0.3679
	c. 0.1120
	d. 0.5783

Question 7

Fill in the blank. Let X₁, X₂, …, X_n be continuous random variables with density functions f₁(x), f₂(x), …, f_n(x). X₁, X₂, … X_n are mutually independent if, and only if, which of the following is true?

Choose one answer.

	a. f(x₁, x₂, …, x_n) = f₁(x₁)f₂(x₂) … f_n(x_n)
	b. f(x₁, x₂, …, x_n) = f₁(x₁)+ f₂(x₂) + … +f_n(x_n)
	c. f(x₁, x₂, … , x_n) = f₁ (x₁ x₂… x_n)
	d. None of the above

Question 8

Fill in the blank. The sample space of an experiment whose outcome depends on chance is the _______________ of all possible outcomes.

Choose one answer.

	a. Size
	b. Sum
	c. Set
	d. Difference

Question 9

Four people, A, B, C, and D, compete for a prize and only one of them can win. The chance that D wins is equal to the chance that either A or B wins. The chance that C wins is equal to the chance that A wins and is twice as much as the chance of B wins. What is the chance that either A or C will win the competition?

Choose one answer.

	a. 25%
	b. 75%
	c. 50%
	d. 10%

Question 10

If the probability of success of a Bernoulli trial is p, what is the probability that the k^th trial (out of k trials) is the first success?

Choose one answer.

	a. P(X = k) = (1 - p)^k
	b. P(X = k) = k! ∙ p ∙ (1 - p)^k-1
	c. P(X = k) = p^k-1 ∙ (1 - p)
	d. P(X = k) = p ∙ (1 - p)^k-1

Question 11

If X and Y are real-valued random variables and c is a constant, which of the following is false?

Choose one answer.

	a. E(X + Y²) = E(X) + E(Y²)
	b. E(X ∙ Y^1/2) = E(X) ∙ E(Y^1/2)
	c. E(X ∙ c) = c E(X)
	d. E(X^1/2) = (E(X)) ^1/2

Question 12

If X and Y are two independent random variables and E(X) and E(Y) are their expected values, respectively, then what is E(X ∙ Y)?

Choose one answer.

	a. E(X) + E(Y)
	b. E(X ) ∙ E(Y)
	c. E(X)/E(Y)
	d. E(X) - E(Y)

Question 13

If X is a random variable with an expected value of 5, what is the expected value of 5 ∙ X?

Choose one answer.

	a. 1
	b. 10
	c. 25
	d. 5

Question 14

On average, how many times does a six-sided die need to be rolled before a 3 turns up?

Choose one answer.

	a. 4
	b. 5
	c. 6
	d. 12

Question 15

Suppose the number of children in a family follows a Poisson distribution with mean μ = 2.2. Find the probability of finding 1 or more children in the family.

Choose one answer.

	a. 0.11
	b. 0.89
	c. 0.19
	d. 0.55

Question 16

Suppose X ~ Uniform(0, 1). What is P(0.2 ≤ X ≤ 0.4)?

Choose one answer.

	a. 0.4
	b. 0.1
	c. 0.2
	d. 0.5

Question 17

The probability of success of a Bernoulli trial is p. Which of the following is the definition of the geometric distribution?

Choose one answer.

	a. The probability distribution of n so that the n^th trial (out of n trials) is the first success.
	b. The number of successes in a sequence of n trials.
	c. The number of successes in a sequence of trials, before a specified (non-random) number r of failures occurs
	d. None of the above

Question 18

What is a Bernoulli trial?

Choose one answer.

	a. An experiment whose outcome is random continuous numbers between 0 and 1
	b. An experiment whose outcome is random integer numbers
	c. An experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure"
	d. An experiment whose outcome is random and can be either of three possible outcomes

Question 19

What is the probability of getting at least one double (e.g. two ones) from two 6-sided dice, after rolling them 5 times?

Choose one answer.

	a. 0.08
	b. 0.40
	c. 0.16
	d. 0.60

Question 20

What is the variance of an exponentially distributed random variable with parameter λ?

Choose one answer.

	a. 1/ λ²
	b. 1/ λ
	c. λ²
	d. λ

Question 21

Which of the following is NOT a continuous probability distribution?

Choose one answer.

	a. Poisson distribution
	b. Gamma distribution
	c. Exponential distribution
	d. Normal distribution

Question 22

Which of the following is NOT a discrete probability distribution?

Choose one answer.

	a. Binomial distribution
	b. Poisson distribution
	c. Exponential distribution
	d. Geometric distribution

Question 23

Which of the following is NOT a discrete random variable?

Choose one answer.

	a. The number of children taller than 4 ft in a family
	b. The number of cars in a family
	c. The number of children in a family
	d. The average height of children in a family

Question 24

Which of the following is the probability density function for an exponential distribution with parameter λ?

Choose one answer.

	a. f(x, λ) = λ e^{- λx}
	b. f(x, λ) = e^{- λx}
	c. f(x, λ) = λ e^{- x}
	d. f(x, λ) =1/ λ e^{- λx}

Question 25

X and Y are real-valued random variables, with expected values of E(X) and E(Y), respectively. What is the expected value of X - Y?

Choose one answer.

	a. E(X) + E(Y)
	b. E(X) - E(Y)
	c. E(X) ∙ E(Y)
	d. X - Y

Question 26

X and Y are real-valued random variables, with variances V(X) and V(Y), respectively. What is the variance of X - Y?

Choose one answer.

	a. V(X/Y)
	b. E(X + Y)²
	c. V(X) + V(Y)
	d. V(X - Y)

Question 27

X and Y are two independent random variables with variances V(X) and V(Y), respectively. C is a constant. Which of the following statement is false?

Choose one answer.

	a. V(C ∙ X) = C²V(X)
	b. V(X + Y) = V(X) + V(Y)
	c. V(X - Y) = V(X) - V(Y)
	d. V(X + C) = V(X)

Question 28

Matt flipped a coin four times. What is the probability that he had exactly two heads?

Choose one answer.

	a. 0.375
	b. 0.25
	c. 0.5
	d. 0.625

Question 29

A coin is biased with a 55% chance of getting tails for every random toss. Estimate the probability of getting exactly 55 heads in 100 tosses of a coin.

Choose one answer.

	a. 0.011
	b. 0.122
	c. 0.082
	d. 0.046

Question 30

A couple plans to have 4 children. They already have one girl. What is the probability that they will have 3 girls in total?

Choose one answer.

	a. 5/8
	b. 1/4
	c. 2/7
	d. 3/8

Question 31

A new test for colorectal cancer in adults is marketed with 99% sensitivity and 95% specificity. That is if a tested patient has colorectal cancer, the test correctly reports "positive result" 99% of the time, and if a tested patient does not have colorectal cancer, the test correctly reports "negative result" 95% of the time. The prevalence of colorectal cancer in adults is 1% (i.e. on average, one in 100 adults has colorectal cancer). What is the probability that an adult has colorectal cancer, if his test returns positive?

Choose one answer.

	a. 99%
	b. 17%
	c. 95%
	d. 13%

Question 32

A new test for colorectal cancer in adults is marketed with 99% sensitivity and 95% specificity. That is if a tested patient has colorectal cancer, the test correctly reports "positive result" 99% of the time, and if a tested patient does not have colorectal cancer, the test correctly reports "negative result" 95% of the time. The prevalence of colorectal cancer in adults is 1% (i.e. on average, one in 100 adults has colorectal cancer). What is the probability that an adult does not have colorectal cancer, if his test returns negative?

Choose one answer.

	a. 99.99%
	b. 95%
	c. 17%
	d. 13%

Question 33

Approximately 25% of smokers have lung cancer. Lung cancer screening by MRI has a sensitivity of 70% and a specificity of 90%. What is the probability that a smoker has lung cancer given that she just had a positive test?

Choose one answer.

	a. 0.77
	b. 0.85
	c. 0.70
	d. 0.55

Question 34

Approximately 25% of smokers have lung cancer. Lung cancer screening by MRI has a sensitivity of 70% and a specificity of 90%. What is the probability a smoker has lung cancer given that she just had a negative test?

Choose one answer.

	a. 0.1
	b. 0.9
	c. 0.17
	d. 0.83

Question 35

Calculate the cumulative distribution function at x = 1 for an exponential distribution with λ = 1.

Choose one answer.

	a. 0.6321
	b. 0.1242
	c. 0.8921
	d. 0.1012

Question 36

Calculate the variance of a geometrically distributed random variable X with parameter p = 0.2.

Choose one answer.

	a. 10
	b. 5
	c. 15
	d. 20

Question 37

Calculate the variance of the following sample: 7, 7, 7, 3, 6, 7, 9, 5, 4, 4.

Choose one answer.

	a. 7.20
	b. 2.67
	c. 4.50
	d. 1.77

Question 38

Estimate the probability of getting exactly 45 heads in 100 tosses of a coin.

Choose one answer.

	a. 0.01
	b. 0.1
	c. 0.08
	d. 0.05

Question 39

Estimate the probability of getting exactly 50 heads in 100 tosses of a coin.

Choose one answer.

	a. 0.5
	b. 0.08
	c. 0.18
	d. 0.04

Question 40

How many times does one need to roll a 6-sided die so that he can be 90% sure that a 6 will turn up?

Choose one answer.

	a. 13
	b. 26
	c. 36
	d. 8

Question 41

How many times does one need to roll a 6-sided die so that he can be 99% sure that a 6 will turn up?

Choose one answer.

	a. 6
	b. 26
	c. 36
	d. 16

Question 42

One in 10 coins has tails on both sides. One coin is chosen at random and flipped three times. What is the probability of getting 3 heads?

Choose one answer.

	a. 0.1725
	b. 0.2125
	c. 0.1125
	d. 0.1250

Question 43

One in 10 coins has tails on both sides. One coin is chosen at random and flipped three times. What is the probability of getting 3 tails?

Choose one answer.

	a. 0.25
	b. 0.2125
	c. 0.0625
	d. 0.1775

Question 44

Suppose that Y ~ Binomial (500, 0.02). Estimate Pr (Y = 16). (Hint: Poisson distribution is a good approximation of the binomial function when p is small).

Choose one answer.

	a. 0.005
	b. 0.12
	c. 0.02
	d. 0.05

Question 45

Suppose the number of cars in a family follows a Poisson distribution with mean μ = 1.5. Find the probability of finding 2 or more cars in the family.

Choose one answer.

	a. 0.11
	b. 0.44
	c. 0.29
	d. 0.35

Question 46

Suppose the number of children in a family follows a Poisson distribution with mean μ = 2.2. Estimate the probability of finding exactly 3 children in the family.

Choose one answer.

	a. 0.087
	b. 0.352
	c. 0.123
	d. 0.197

Question 47

Suppose X~Normal(140, 20). Calculate P(X ≥ 200).

Choose one answer.

	a. 1.3%
	b. 2.25%
	c. 0.78%
	d. 0.13%

Question 48

The average GPA of students at a university is 3.0 with a standard deviation of 1.0. A random sample of 179 students is collected. What is the probability that the average GPA of this sample is higher than 3.0?

Choose one answer.

	a. 60%
	b. 50%
	c. 75%
	d. 65%

Question 49

The average height of men in the U.S. is 1.8 m with a standard deviation of 8 cm. A random group of 16 men is selected. What is the probability that the average height of this group is higher than 1.84 m?

Choose one answer.

	a. 0.0228
	b. 0.0456
	c. 0.0114
	d. 0.0765

Question 50

The average height of men in the U.S. is 1.8 m with a standard deviation of 8 cm. What is the percentage of men shorter than 1.88 m and taller than 1.72 m?

Choose one answer.

	a. 57%
	b. 68%
	c. 32%
	d. 43%

Question 51

The probably of carrying a cancer mutation in the general population is 1 in 1000. What is the probability of finding 3 people with the mutation in a random sample of 500 people? (Hint: Poisson distribution is a good approximation of the binomial function when p is small).

Choose one answer.

	a. 0.0126
	b. 0.0523
	c. 0.0472
	d. 0.0052

Question 52

The probably of carrying a cancer mutation in the general population is 1 in 1000. What is the probability of finding no one with the mutation in a random sample of 500 people?

Choose one answer.

	a. 0.60
	b. 0.25
	c. 0.82
	d. 0.50

Question 53

X is a random variable with a variance of 1. What is the variance of 5 ∙ X?

Choose one answer.

	a. 0.04
	b. 0.2
	c. 5
	d. 25

Question 54

X is a random variable with a variance of 1. What is the variance of X + 2?

Choose one answer.

	a. 4
	b. 2
	c. 1
	d. 0.5

Question 55

The average GPA of students at a university is 3.0 with a standard deviation of 1.0. A random sample of 9 students is collected. What is the probability that the average GPA of this sample is higher than 3.3?

Choose one answer.

	a. 0.231
	b. 0.821
	c. 0.184
	d. 0.124