University:
Stony Brook University
Course:
AMS 310 | Survey of Probability and Statistics
Academic year:

2024
Views:

356

Pages:

3
Author:

study_genius

AMS 310 More Test 2 Practice Problems Solutions Fred Rispoli 1. Let X be a random variable with pdf f(x) = kx, 0 < x < 4. a) Find the value of k. show all steps. 4 1 ∫0 𝑘𝑥 𝑑𝑥 = 1 implies k = 8 b) Find the cdf of X 𝑥1 1 F(x) = ∫0 8 𝑡 𝑑𝑡 = 16 𝑥 2 c) Find P(X < 1 or X > 3) 32 1 8 P(X < 1 or X > 3) = F(1) + (1 – F(3) ) = 16 + (1 − 16) = 16 = 0.5 d) Find E(X) 41 64 8 E(X) = ∫0 8 𝑥 2 𝑑𝑥 = 24 = 3 = 2.67 2. The time it takes a mechanic to change oil has an exponential distribution with mean 20. a) Set up an integral to find P(15 < X < 25), then evaluate the integral. 25 1 P(15 < X < 25) = ∫15 20 𝑒 −𝑥/20 𝑑𝑥 = 𝑒 −5/4 − 𝑒 −3/4 = 0.18 b) Find the 40th percentile Solve F(X) = 0.4 x  10.2 1 - 𝑒 −𝑥/20 = 0.4 3. The random variables X and Y have the following joint probability distribution. X Y f(x,y) 1 2 1 0.10 0.20 2 0.30 0.15 3 0.20 0.05 This study source was downloaded by 100000819498514 from CourseHero.com on 11-07-2024 09:40:31 GMT -06:00 https://www.coursehero.com/file/31571602/AMS-310-More-Test-2-Practice-Problemspdf/ a) Find 𝑃(𝑋 + 𝑌 > 3). 𝑃(𝑋 + 𝑌 > 3) = 0.20 + 0.15 + 0.05 = 0.40 b) Find the marginal probability distributions 𝑓1 (𝑥) and 𝑓2 (𝑦). X f1(X) 1 2 3 Y 1 2 0.3 0.45 0.25 f2(y) 0.6 0.4 c) Determine if X and Y are independent. Justify your answer. Not independent f(1,1) = 0.10, but f1(1)f2(1) = 0.3(0.6) = 0.18 4. Suppose the random variables X and Y have joint pdf as follows: 𝑐𝑥 + 1 𝑥, 𝑦 ≥ 0 𝑎𝑛𝑑 𝑥 + 𝑦 < 1 f(x,y) = { 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 a) Find the constant c 1 1−𝑥 ∫0 ∫0 1 𝑐 𝑐 𝑐𝑥 + 1 𝑑𝑦 𝑑𝑥 = ∫0 𝑐𝑥(1 − 𝑥) + (1 − 𝑥) 𝑑𝑥 = 2 − 3 + 1 − 1 2 = 1. This implies that c = 3 b) Find the marginal pdf f1(x) of X 1−𝑥 ∫0 3𝑥 + 1 𝑑𝑦 = [3xy + y] 𝑦=1−𝑥 = -3x2 + 2x + 1 = (3x + 1)(1 – x), 0 ≤ x ≤ 1 𝑦=0 1 1 c) Find P(X < 2, Y < 2 ) 0.5 0.5 0.5 = ∫0 ∫0 (3𝑥 + 1)𝑑𝑦 𝑑𝑥 = ∫0 (1.5𝑥 + 0.5) 𝑑𝑥 = 0.4375 5. Let the random variable have joint pdf as follows: f(x, y) = 3 4 (𝑥 2 + 3𝑦 2 ) 0 < x < 1, 0 < y < 1 This study source was downloaded by 100000819498514 from CourseHero.com on 11-07-2024 09:40:31 GMT -06:00 https://www.coursehero.com/file/31571602/AMS-310-More-Test-2-Practice-Problemspdf/ a) Find the marginal densities of X and Y 1 f1(x) = ∫0 𝑓(𝑥, 𝑦)𝑑𝑦 = 1 f2(y) = ∫0 𝑓(𝑥, 𝑦)𝑑𝑥 = 3 4 1 4 (𝑥 2 + 1), for 0 < x < 1 (1 + 9𝑦 2 ), for 0 < y < 1 b) Determine the conditional pdf’s f1(x | y) and f2(y| x) f1(x | y) = f2(y | x) = 𝑓(𝑥,𝑦) 𝑓2 (𝑦) 𝑓(𝑥,𝑦) 𝑓1 (𝑥) = 3 2 (𝑥 +3𝑦 2 ) 4 1 (1+9𝑦 2 ) 4 3 (𝑥 2 +3𝑦 2 ) = 43 4 (𝑥 2 +1) = = 3𝑥 2 +9𝑦2 1+9𝑦 2 𝑥 2 +3𝑦2 𝑥 2 +1 c) Find E(X) and E(Y) 1 9 1 11 16 E(X) = ∫0 𝑥𝑓1 (𝑥)𝑑𝑥 = 16 E(Y) = ∫0 𝑦𝑓2 (𝑦)𝑑𝑦 = 6. If the variance of a normal population is 9, what is the probability that the variance of a random sample of size 12 exceeds 16.102? a) Find the probability using the distribution table. b) Not on test but just for fun, find the probability using R or Excel 𝜒2 = (𝑛 − 1)𝑆 2 11𝑆 2 = is distributed as 𝜒 2 with 11 degrees of freedom. 𝜎2 9 (a) 11𝑆 2 11(16.102 𝑃(𝑆 > 16.102) = 𝑃 ( > ) = 𝑃(𝜒 2 > 19.68) = 0.05 9 9 2 (b) >1-pchisq(19.68, 11) = 0.05 This study source was downloaded by 100000819498514 from CourseHero.com on 11-07-2024 09:40:31 GMT -06:00 https://www.coursehero.com/file/31571602/AMS-310-More-Test-2-Practice-Problemspdf/ Powered by TCPDF (www.tcpdf.org)