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1. Suppose that X1 and X2 are independent random variables having the uniform dis- tribution on (0, 1, 2, 3, 4, 5). That is, S) = P{X; = j} = for i = 1, 2 and 0

1. Suppose that X1 and X2 are independent random variables having the uniform dis- tribution on (0, 1, 2, 3, 4, 5). That is, S) = P{X; = j} = for i = 1, 2 and 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 20E
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1. Suppose that X1 and X2 are independent random variables having the uniform dis- tribution on (0, 1, 2, 3, 4, 5). That is, S) = P{X; = j} = for i = 1, 2 and 0 <j<5. a. Compute E(X1) and Var(X1). [Hint: Use the formulas E(X) = EiR), Var(X) = EA) – (E(X))’.] b. Determine the probability distribution for Y = X1 + X2. [Hint: For each possible value of Y, determine all combinations of X1 and X, that result in that value. For example, Y = 3 can be obtained by (X1, X2) = (0, 3), (1, 2), (2, 1), and (3, 0). Since each pair has probability ()() = 36, we obtain P{Y = 3} = =5. Repeat this process for all values of Y. As a check be sure that EP{Y = y] = 1.0.] ve Quality and Assurance c. Using the results of part (b), find P{1.5 < Y< 6.5}. d. Using the results E(Y) = 2E(X1) and Var(Y) = 2Var(X1), approximate the an- swer to part (c) using a normal distribution. e. Suppose that X1, X2, . .., X20 are independent identically distributed random variables having the uniform distribution on (0, 1, 2, 3, 4, 5). Using a normal ap- proximation, estimate 20 2 X; s 75 f. Do you think that the approximation computed in part (d) or part (e) is more ac- curate? Why?
Transcribed Image Text:1. Suppose that X1 and X2 are independent random variables having the uniform dis- tribution on (0, 1, 2, 3, 4, 5). That is, S) = P{X; = j} = for i = 1, 2 and 0 <j<5. a. Compute E(X1) and Var(X1). [Hint: Use the formulas E(X) = EiR), Var(X) = EA) – (E(X))’.] b. Determine the probability distribution for Y = X1 + X2. [Hint: For each possible value of Y, determine all combinations of X1 and X, that result in that value. For example, Y = 3 can be obtained by (X1, X2) = (0, 3), (1, 2), (2, 1), and (3, 0). Since each pair has probability ()() = 36, we obtain P{Y = 3} = =5. Repeat this process for all values of Y. As a check be sure that EP{Y = y] = 1.0.] ve Quality and Assurance c. Using the results of part (b), find P{1.5 < Y< 6.5}. d. Using the results E(Y) = 2E(X1) and Var(Y) = 2Var(X1), approximate the an- swer to part (c) using a normal distribution. e. Suppose that X1, X2, . .., X20 are independent identically distributed random variables having the uniform distribution on (0, 1, 2, 3, 4, 5). Using a normal ap- proximation, estimate 20 2 X; s 75 f. Do you think that the approximation computed in part (d) or part (e) is more ac- curate? Why?
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