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
A First Course in Probability (10th Edition)
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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?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa11c0cf6-9649-4556-9330-71e063d0199c%2Ff91d1130-6b44-494f-ab17-2032be02d4f9%2F5fsue4.png&w=3840&q=75)
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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