A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 10, Problem 10.9P

Suppose we have a method for simulating random variables from the distributions F1 and F2. Explain how to simulate from the distribution F ( x ) = p F 1 ( x ) + ( 1 p ) F 2 ( x ) 0 < p < 1

Give a method for simulating from F ( x ) = { 1 3 ( 1 e 3 x ) + 2 3 x 0 < x 1 4 1 3 ( 1 e 3 x ) + 2 3 x > 1

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Suppose X and Y are independent. X has a mean of 1 and variance of 1, Y has a mean of 0, and variance of 2. Let S=X+Y, calculate E(S) and Var(S). Let Z=2Y^2+1/2 X+1 calculate E(Z). Hint: for any random variable X, we have Var(X)=E(X-E(X))^2=E(X^2 )-(E(X))^2, you may want to find EY^2  with this. Calculate cov(S,X). Hint: similarly, we have cov(Z,X)=E(ZX)-E(Z)E(X), Calculate cov(Z,X). Are Z and X independent? Are Z and Y independent? Why? What about mean independence?
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