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

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

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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?

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

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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?

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

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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?

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

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

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

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

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

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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?

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

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

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

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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?

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

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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?

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

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

	a. E(X + Y2) = E(X) + E(Y2)
	b. E(X ∙ Y1/2) = E(X) ∙ E(Y1/2)
	c. E(X ∙ c) = c E(X)
	d. E(X1/2) = (E(X)) 1/2

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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)?

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

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

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

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

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

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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.

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

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

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

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

	a. The probability distribution of n so that the nth 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

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What is a Bernoulli trial?

	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

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

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

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

	a. 1/ λ2
	b. 1/ λ
	c. λ2
	d. λ

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Which of the following is NOT a continuous probability distribution?

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

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Which of the following is NOT a discrete probability distribution?

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

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Which of the following is NOT a discrete random variable?

	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

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

	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

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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?

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

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

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

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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?

	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)

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

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

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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.

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

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

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

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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?

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

.

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?

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

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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?

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

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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?

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

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

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

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

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

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

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

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

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

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

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

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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?

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

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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?

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

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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?

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

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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?

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

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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).

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

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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.

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

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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.

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

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

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

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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?

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

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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?

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

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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?

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

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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).

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

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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?

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

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

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

.

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

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

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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?

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

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