1) Assume a relationship Y = g2(91(X)) between the two random variables X and Y where gi and g2 are two functions. To compute the mean of Y, we can choose any of the following equivalent expressions: E(Y) = | fr (u)ydy = | fn(x)(u)g2(u)du = / fx(#)g2(9n(x)dr (1) -00 where fy (y), f1(x)(u) and fx(x) are the probability density functions (p.d.f.s) of Y, 91(X) and X, respectively. Now consider Y = X? where X is N(0, o2). a) Determine the p.d.f. fy (y) of Y. Hint: for y > 0, fy (y) Prob(Y < y) = Prob(-< X < VI) = % x (x)dx. b) Then evaluate explicitly E(Y)= S fy(y)ydy.
1) Assume a relationship Y = g2(91(X)) between the two random variables X and Y where gi and g2 are two functions. To compute the mean of Y, we can choose any of the following equivalent expressions: E(Y) = | fr (u)ydy = | fn(x)(u)g2(u)du = / fx(#)g2(9n(x)dr (1) -00 where fy (y), f1(x)(u) and fx(x) are the probability density functions (p.d.f.s) of Y, 91(X) and X, respectively. Now consider Y = X? where X is N(0, o2). a) Determine the p.d.f. fy (y) of Y. Hint: for y > 0, fy (y) Prob(Y < y) = Prob(-< X < VI) = % x (x)dx. b) Then evaluate explicitly E(Y)= S fy(y)ydy.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:1) Assume a relationship Y = g2(91(X)) between the two random variables X and Y where
gi and g2 are two functions. To compute the mean of Y, we can choose any of the following
equivalent expressions:
E(Y)=
| fr (u)ydy = | fn(x)(u)g2(u)du = / fx(#)g2(9n(x)dr
(1)
-00
where fy (y), f91(x)(u) and fx(x) are the probability density functions (p.d.f.s) of Y,
91(X) and X, respectively. Now consider Y = X? where X is N(0, o2).
a) Determine the p.d.f. fy (y) of Y. Hint: for y > 0, fy (y)
Prob(Y < y) =
Prob(-< X < VI) = % x (x)dx.
b) Then evaluate explicitly E(Y)= S fy(y)ydy.
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