Draw Two Cards From a Standard Deck of 52 Playing Cards Successively

Practice Final Exam A. Draw two cards from a standard deck of 52 playing cards successively. (1) What is the conditional probability that the second card is a king given that the first card is a king? (a) 0.059 (b) 0.078 (c) 0.053 (d) 0.091 (2) What is the conditional probability that the second card is a king given that the first card is a diamond? (a) 0.059 (b) 0.077 (c) 0.250 (d) 0.333 (3) What is the probability that at least one card is a king? (a) 0.059 (b) 0.079 (c) 0.149 (d) 0.250 (4) What is the probability that exactly one card is a king? (a) 0.059 (b) 0.145 (c) 0.153 (d) 0.250 (5) Given that at least one card is a king, what is the conditional probability that both cards are kings? (a) 0.009 (b) 0.013 (c) 0.027 (d) 0.059 (6) Given that at least one card is a king, what is the conditional probability that at least one card is a diamond? (a) 0.250 (b) 0.333 (c) 0.389 (d) 0.443 B. Let A and B be two events such that P (A) > 0 and P (B) > 0. (7) If we know that A′ ∩ B = A ∩ B ′ , then what must be true? (a) P (A) < P (B) (c) P (A | B) = P (B | A) (b) P (A ∩ B) = 0 (d) P (A) > P (B ) (8) Suppose that P (A) = 0.4, P (A ∪ B ) = 0.8, and P (A ∩ B ) = 0.2. Let C denote the event that exactly one of A and B occurs. Then P (C) is equal to (a) 0.2 (b) 0.4 (c) 0.6 (d) 0.8 (9) Suppose that P (A | B) = 1. Then P (B ′ | A′ ) is equal to (a) 0 (b) 0.25 (c) 0.5 (d) 1.0 (10) Suppose that P (A) = 0.3, P (B ) = 0.5, and P (A′ ∪ B ) = 0.8. Then P (A ∩ B) is equal to (a) 0.0 (b) 0.1 (c) 0.2 (d) 0.4 C. Let X be a random variable with the pdf f (x) = x2 /c for x = −2, −1, 1, 2 and some constant c. (11) What statement is correct? (a) X is a continuous random variable (c) support of X is S = [−2, 2] (b) X is a discrete random variable (d) support of X is S = {2, −1, 0, 1, 2} (12) The constant c is equal to (a) 1 (b) 5 (c) 8 (d) 10 (13) The mean of X is equal to (a) -1 (b) 0 (c) 1 (d) 2 (14) E[X] is equal to (a) -2 (b) -1 (c) 0 (d) 1 (15) The variance of X is equal to (a) 3/5 (b) 7/16 (c) 17/5 (d) 26/7 (16) E[X 3 ] is equal to (a) -16 (b) -5 (c) 0 (d) 18 D. Suppose that we roll two fair six-sided dice simultaneously. (17) What is the probability that the sum of numbers on two dice is even? (a) 1/4 (b) 1/3 (c) 1/2 (d) 3/4 (18) What is the probability that the sum of numbers is at most 11? (a) 1/36 (b) 1/6 (c) 15/36 (d) 35/36 (19) What is the probability that the maximal number among two numbers is at least 5? (a) 1/9 (b) 7/36 (c) 5/9 (d) 31/36 (20) What is the probability that the minimal number among two numbers is at most 2? (a) 1/9 (b) 7/36 (c) 5/9 (d) 31/36 E. It is believed that 20% of Americans do not have any health insurance. Suppose this is true and let X equal the number of people with no health insurance in a sample of n Americans. (21) What is the distribution of X (a) Normal (b) Uniform (c) Binomial (d) Poisson (22) If n = 25, then the variance of X is equal to (a) 2 (b) 4 (c) 8 (d) 16 (23) If n = 10, then P (X ≥ 2) is equal to (a) 0.333 (b) 0.625 (c) 0.737 (d) 0.956 (24) If n = 25, then P (X ≥ 10) is approximately equal to (a) 0.0062 (b) 0.0136 (c) 0.0274 (d) 0.1435 F. Let X be a random variable denoting the number of calls received by the customer service of an insurance company on a given day. Suppose that X has mean 4. (25) What is the distribution of X ? (b) Bernoulli (a) Normal (c) Binomial (d) Poisson (26) What is the variance of X ? (a) 1 (b) 2 (c) 3 (d) 4 (27) What is P (X ≤ 2)? (a) 0.1 (b) 0.25 (c) 0.5 (d) 0.75 (c) 0.55 (d) 0.85 (28) What is P (X > 3)? (a) 0.1 (b) 0.4 (29) What is E[X 2 ]? (a) 1 (b) 16 (c) 20 (d) 40 G. Let X1 and X2 be random variables with support S1 = {0, 1} and S2 = {−1, 1}, respectively, and with the joint pdf f (x1 , x2 ) such that f (0, −1) = 1/3, f (0, 1) = 1/3, f (1, −1) = 1/6 and f (1, 1) = 1/6. (30) Which claim is correct? (a) X1 and X2 are independent (c) X1 and X2 are mutually exclusive (b) X1 and X2 are not independent (d) X1 and X2 are not mutually exclusive (31) The mean of X1 is equal to (a) 0 (b) 1/6 (c) 1/3 (d) 1/2 (32) The variance of X1 is equal to (a) 1/9 (b) 2/9 (c) 1/3 (d) 2/3 (33) The mean of X2 is equal to (a) -1 (b) 0 (c) 1 (d) 2 (34) The variance of X2 is equal to (a) -1 (b) 0 (c) 1 (d) 2 (35) The covariance between X1 and X2 is equal to (a) -1 (b) 0 (c) 1 (d) 2 (36) The correlation between X1 and X2 is equal to (a) -0.5 (b) -0.2 (c) 0.1 (d) 0 H. Let X be a random variable with pdf f (x) = (x − 1)2 /3, x ∈ [0, c], where c is some constant. (37) The constant c is equal to (a) 1 (b) 2 (c) 3 (d) 4 (38) The mean of X is equal to (a) 1/4 (b) 3/4 (c) 7/4 (d) 9/4 (39) The expected value of X 2 /2 is equal to (a) 2.85 (b) 3.67 (c) 4.81 (d) 5.11 (c) 0.81 (d) 1.00 (40) The variance of X is equal to (a) 0.16 (b) 0.64 I. Let X and Y be independent random variables. Suppose that X has mean 1 and variance 4. Also, suppose that X + Y has mean 3 and variance 13. (41) The mean of Y is equal to (a) 1 (b) 2 (c) 3 (d) 4 (42) The standard deviation of Y is equal to (a) 1 (b) 2 (c) 3 (d) 4 (43) The mean of X 2 + Y 2 /2 is equal to (a) 0.15 (b) 3.8 (c) 7.9 (d) 11.5 (44) The variance of X − Y is equal to (a) -5 (b) 5 (c) 9 (d) 13 J. Let X1 , . . . , X5 be a random sample from N (µ, σ2 ) distribution. Suppose that X1 = 1, X2 = 1, X3 = 0, X4 = 2, and X5 = 3. (45) The sample mean is equal to (a) 0 (b) 1.0 (c) 1.4 (d) 1.8 (46) The sample variance is equal to (a) 1 (b) 1.3 (c) 1.6 (d) 1.9 (47) If σ 2 = 1, then the 95% confidence interval for µ is (a) [0.55; 2.25] (b) [0; 2.8] (c) [1.0; 1.8] (d) [0.4; 2.4] (48) If σ 2 is unknown, then 90% confidence interval for µ is (a) [0.55; 2.25] (b) [0; 2.8] (c) [1.0; 1.8] (d) [0.3; 2.5] K. Let X1 , . . . , Xn be a random sample from a continuous distribution with mean µ = 1 and variance σ 2 = 9. (49) Suppose that n = 4. What must be true? (b) P (−4 < X1 < 6) < 16/25 (d) P (−4 < X1 < 1) ≥ 16/25 (a) P (−4 < X1 < 6) ≥ 16/25 (c) P (1 < X1 < 6) < 16/25 (50) Suppose that n = 100. The sample mean based on the random sample X1 , . . . , Xn with high probability is close to (a) 0 (b) 1 (c) 3 (d) 9 (51) Suppose that n = 100. Calculate approximately the probability that the sample mean based on the random sample X1 , . . . , Xn is larger than 1.3. (a) 0.16 (b) 0.4 (c) 0.6 (d) 0.84 (52) Suppose that n = 100. Calculate approximately the probability that the sample mean based on the random sample X1 , . . . , Xn is larger than 0.7. (a) 0.16 (b) 0.4 (c) 0.6 (d) 0.84 (53) Suppose that n = 50. Calculate approximately the probability that the sample mean based on the random sample X1 , . . . , Xn is equal to one. (a) 0 (b) 0.1 (c) 0.5 (d) 1.0 L. Let X1 , . . . , Xn be a random sample from N (µ, σ2 ) distribution of size n = 25. Suppose that we want to test H0 : µ = 0 against H1 : µ 6= 0. (54) Suppose that σ 2 = 1. Find the probability of type 1 error of the test that accepts H0 if −0.3 ≤ ¯Xn ≤ 0.3 and accepts H1 otherwise. (a) 0.14 (b) 0.4 (c) 0.6 (d) 0.86 (55) Suppose that σ 2 is unknown. Find the probability of type 1 error of the test that accepts H0 if −0.137 ≤ X̄n /S ≤ 0.137. (a) 0.5 (b) 0.6 (c) 0.7 (d) 0.8 M. Let X1 , . . . , Xn be a random sample from N (µ, σ2 ) distribution. (56) Suppose that we want to test H0 : µ = 2 against H1 : µ > 2 and n = 100. If σ 2 = 9, then the test of significance level 5% accepts H0 if the sample mean belongs to the interval (a) (−∞, 1.5] (b) (−∞, 2.5] (c) (−2.5, ∞) (d) (1.5, ∞) (57) Suppose that we want to test H0 : µ = 0 against H1 : µ < 0 and n = 20. If σ 2 is unknown, then the test of significance level 5% accepts H0 if the sample mean belongs to the interval (a) [−0.38S, ∞) (b) (−∞, 0.71S] (c) (−∞, −0.38S] (d) [0.71S, ∞) N. Consider a random sample X1 , . . . , Xn of Bernoulli trials with the success probability p = 0.3. Let X̄n be the sample mean. (58) If n = 6, the probability P ( ¯Xn ≤ 1/6) is (a) 0.28 (b) 0.67 (c) 0.42 (d) 0.99 (59) If n = 6, the probability P ( ¯Xn ≤ 5/6) is (a) 0.812 (b) 0.864 (c) 0.916 (d) 0.999 (60) The variance σ 2 of Xi is (a) 0.30 (b) 0.21 (c) 0.70 (d) 1 (61) If n = 100, the probability P ( ¯Xn ≤ 0.35) approximately is (a) 0.61 (b) 0.50 (c) 0.86 (d) 0.98 (62) If n = 100, the probability P ( ¯Xn ≤ 0.25) approximately is (a) 0.37 (b) 0.50 (c) 0.14 (d) 0.61 O. Let X1 and X2 be two random variables, and let Y = (X1 + X2 )2 . Suppose that E[Y ] = 25 and that the variance of X1 and X2 are 9 and 16, respectively. (63) Suppose that both X1 and X2 have mean zero. Then the correlation between X1 and X2 is equal to (a) -1 (b) -0.5 (c) 0 (d) 0.5 (64) Suppose that both X1 and X2 have mean one. Then the covariance between X1 and X2 is equal to (a) -2 (b) -1 (c) 0 (d) 1 (65) Suppose that both X1 and X2 have mean one. Then the correlation between X1 and X2 is equal to (a) -1/6 (b) 0 (c) 1/6 (d) 1/2 P. Let X be a random variable with the cdf F (x) = (x − 1)2 if x ∈ [1, 2]. Let f (x) be the pdf of the random variable X . (66) What is the value of F (0)? (a) 0 (b) 0.3 (c) 0.5 (d) 1 (67) What is the value of F (3)? (a) 0 (b) 0.3 (c) 1.0 (d) 4.0 (68) What is the value of f (1.5). (a) 0 (b) 0.3 (c) 0.5 (d) 1.0 (69) The probability P (X > 1.5) is equal to (a) 0.25 (b) 0.5 (c) 0.75 (d) 0.9 (70) The probability P (X = 1.3) is equal to (a) 0 (b) 0.3 (c) 0.6 (d) 1