Math 180A (P. Fitzsimmons) Final Exam Solutions Each problem is worth 10 points. 1. Suppose that A and B are events with P[B c ] = 7/12, P[B \ A] = 1/6, and P[A \ B] = 1/3. (a) Find P[A]. (b) Find P[A ∪ B]. (Recall that B \ A denotes the event B ∩ Ac , etc.) Solution. First observe that P[B] = 1 − P[B c ] = 1 − 7/12 = 5/12. Therefore, because 5/12 = P[B] = P[AB] + P[B \ A] = P[AB] + 1/6, we have P[AB] = 5/12 − 1/6 = 3/12. (a) P[A] = P[AB] + P[A \ B] = 3/12 + 1/3 = 7/12. (b) P[A ∪ B] = P[A] + P[B \ A] = 7/12 + 1/6 = 9/12 = 3/4. [Alternatively, P[A ∪ B] = P[B] + P[A \ B] = 5/12 + 1/3 = 9/12 = 3/4.] 2. A bowl contains 5 red marbles and 3 green marbles. Marbles are drawn at random from the bowl, without replacement. Let Rk be the event that the marble chosen on draw k is red, and let Gk be the event that the marble chosen on draw k is green. (a) Find the probability that the first red marble to be drawn is the third marble drawn. (b) Find P[R3 |G1 ]. Solution. (a) This is the same as the probability of G1 G2 R3 , which is P[G1 G2 R3 ] = P[G1 ]P[G2 |G1 ]P[R3 |G1 G2 ] = 5 3 2 5 · · = = 0.08928. 8 7 6 56 (b) First, P[G1 R3 ] = P[G1 G2 R3 ] + P[G1 R2 R3 ] 5 + P[G1 ]P[R2 |G1 ]P[R3 |G1 R2 ] = 56 5 3 5 4 = + · · 56 8 7 6 15 . = 56 Consequently P[R3 |G1 ] = 15 . 3 5 = = 0.71428. 56 8 7 3. The joint probabilities for a pair of discrete random variables are given by the formula ( x(x + y) , x = 1, 2, 3, y = 1, 2, 3; P(X = x, Y = y) = 78 0, otherwise. (a) Find the marginal probabilities P(X = x) and P(Y = y) for x, y = 1, 2, 3. 1 (b) Find E[X]. (c) Find E[X|Y = 2]. [You may find it helpful to display the joint probability function in the form of a table.] Solution. (a) The marginal probabilities for X are P[X = 1] = 3 X P[X = 1, Y = j] = 2+3+4 9 = 78 78 P[X = 2, Y = j] = 24 6 + 8 + 10 = 78 78 P[X = 3, Y = j] = 12 + 15 + 18 45 = . 78 78 j=1 P[X = 2] = 3 X j=1 P[X = 3] = 3 X j=1 The marginal probabilities for X are P[Y = 1] = 3 X P[X = j, Y = 1] = 2 + 6 + 12 20 = 78 78 P[X = j, Y = 2] = 26 3 + 8 + 15 = 78 78 P[X = j, Y = 3] = 4 + 10 + 18 32 = . 78 78 j=1 P[Y = 2] = 3 X j=1 P[Y = 3] = 3 X j=1 (b) Using (a) E[X] = 1 · P[X = 1] + 2 · P[X = 2] + 3 · P[X = 3] 9 24 45 =1· +2· +3· 78 78 78 9 + 48 + 135 192 32 = = = 78 78 13 = 2.46. (c) Evidently 3 26 8 P[X = 2|Y = 2] = 26 15 P[X = 3|Y = 2] = 26 P[X = 1|Y = 2] = Therefore E[X|Y = 2] = 1 · 8 15 64 32 3 +2· +3· = = = 2.46. 26 26 26 26 13 4. A fair six-sided die is tossed repeatedly. Let N be the number of different faces that appear in the first eight rolls. Calculate E(N ). [Hint: Indicator random variables.] 2 Solution. Let Fk be the event that face k shows up at least once in the eight tosses. Since the complement of Fk is the event that face k is never tossed, P[Fk ] = 1 − (5/6)8 , k = 1, 2, 3, 4, 5, 6. As hinted, N = IF1 + IF2 + · · · IF6 , and so E[N ] = E[IF1 ] + · · · E[IF6 ] = 6P[F1 ] = 1(1 − (5/6)8 ) = 4.6046. 5. One thousand cards are drawn (with replacement) from a standard deck of 52 cards. Let X be the total number of Kings drawn. (a) Find E[X] and Var[X]. (b) Use the normal approximation to compute P[65 ≤ X ≤ 90]. Solution. (a) The random variable X has the binomial distribution with parameters n = 1000 and p = 1/13. Therefore µ = E[X] = 1000/13 = 76.92 and Var[X] = 1000(1/13)(12/13) = 71.01. The √ associated standard deviation is σ = 71.01 = 8.43. (b) Using the normal approximation to the binomial (including the continuity correction): 64.5 − 76.92 90.5 − 76.92 −Φ P[65 ≤ X ≤ 90] = Φ 8.43 8.43 = Φ(1.61) − Φ(−1.47) = Φ(1.61) − 1 + Φ(1.47) = .9463 − 1 + .9292 = .8755. 6. Customers arrive at Bob’s Bargain Bookstore in accordance with a Poisson process of intensity 10 customers per hour. The store opens at 8 am, but Bob doesn’t arrive until 10 am. (a) What is the average number of customers that arrive before Bob gets to his store (that is, between 8 am and 10 am)? (b) What is the probability that no customer arrives before Bob? (c) Let T be the amount of time (in hours) that Bob waits until he sees a customer walk in the door. Find P[T > x] for x > 0. Solution. Let N (s, t] denote the number of customers arriving in the time interval (s, t] (with time being measured in hours after opening). We know that N (s, t] has the Poisson distribution with expectation 10 · (t − s). (a) This is E[N (0, 2]] = 10 · (2 − 0) = 20. (b) P[N (0, 2] = 0] = exp(−20) = 2 × 109 . ((c) P[T > x] = P[N (2, 2 + x] = 0] = e−10x , for x > 0. (In other words, T has the exponential distribution with parameter 10.) 3 7. A random variable X has the continuous-type density function 3 − x2 ), 0 < x < 1; 0, otherwise. 2 (1 f (x) = (a) Find P[1/3 < X ≤ 2/3]. (b) Find E[X]. (c) Find Var[X]. Solution. (a) Z 2/3 3 (1 − x2 ) dx 1/3 2 2/3 3 = (x − x3 /3) 2 1/3 3 2 8/27 1 1/27 = − − − 2 3 3 3 3 10 = 0.3704. = 27 P[1/3 < X ≤ 2/3] = (b) Z 1 3 3 E[X] = x · (1 − x2 ) dx = 2 2 0 3 3 1 1 − = . = 2 2 4 8 1 Z (x − x3 ) dx 0 (c) First, Z 1 3 3 x2 · (1 − x2 ) dx = 2 2 0 3 1 1 1 = − = . 2 3 5 5 E[X 2 ] = Consequently, Var[X] = (1/5) − (3/8)2 = 19 320 Z 1 (x2 − x4 ) dx 0 = 0.059375. 8. Suppose that the random variable X has the exponential distribution with parameter 2. Let Y = X 3. (a) Find the density function fY for Y . (b) Find the cumulative distribution function FY for Y . Solution. (a) We are told that fX (x) = 2e−2x for x > 0. Setting y = x3 , we have x = y 1/3 and dx/dy = 3y12/3 . Therefore fY (y) = fX (y 1/3 ) · 1/3 1 2 = 2/3 e−2y , 2/3 3y 3y 4 y > 0. (b) Integrating, we find FY (y) = P[Y ≤ y] Z y = fY (t) dt 0 Z y 2 −2t1/3 = e dt. 2/3 0 3t Making the change of variables u = t1/3 (so that t = u3 and dt = 3u2 du) this integral becomes Z y 1/3 2e−2u du = (1 − e−2y 1/3 ), 0 for y > 0. Alternatively FY (y) = P[Y ≤ y] = P[X 3 ≤ y] = P[X ≤ y 1/3 ] = FX (y 1/3 ) = (1 − e−2y 1/3 ), for y > 0. 9. A random point (X, Y ) is chosen from unit square with continuous-type joint density function c(3x2 + 2y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1; f (x, y) = 0, otherwise. (a) Find c. (b) Find the marginal density function fY for Y . (c) Find E(Y ). Solution. (a) The double integral of the density must be equal to 1; therefore 1 Z Z 1=c 1 1 Z 2 (3x + 2y) dx dy = 0 0 (1 + 2y) dy = 1 + 1 = 2, 0 which means that c = 1/2. (b) Integrate: 1 Z fY (y) = f (x, y) dx = 0 1 Z 1 2 (3x2 + 2y) dx 0 1 1 = (1 + 2y) = + y, 2 2 for 0 < y < 1. (c) Z 1 E[Y ] = Z yfY (y) dy = 0 Z = (y/2 + y 2 ) dy 0 = y(1/2 + y) dy ) 1 1 1 7 + = . 4 3 12 5 1 10. Suppose X and Y are independent random variables such that X has uniform (0, 1) distribution, and Y has exponential distribution with mean 1. Calculate: (a) E(X + Y ); (b) E(XY ); (c) E (X − Y )2 . endproclaim R1 Solution. Observe that E[X] = 1/2 and E[Y ] = 1. Also, E[X 2 ] = 0 x2 dx = 1/3 and E[Y 2 ] = Var[Y ] + (E[Y ])2 = 1 + 12 = 2. (a) E[X + Y ] = E[X] + E[Y ] = 1/2 + 1 = 3/2. (b) E[XY ] = E[X] · E[Y ] = (1/2) · 1 = 1/2. ((c) E (X − Y )2 = E[X 2 − 2XY + Y 2 ] = E[X 2 ] − 2E[XY ] + E[Y 2 ] = 1/3 − 2(1/2) + 2 = 4/3. 6
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