Exercise I. Let X,Y~ N(0, 1) be jointly Gaussian random variables and let t€ R. The goal of this exercise is to prove the following identity, using three different ways: E[Xe] =tE[XY]E[e¹Y]. (*) 1. a. Prove that E[Xe¹x] = tE[e¹x]. b. Set p = E[XY]. Show that (X,Y) law (X, pX+√√1 - p²Z), where Z~ N(0, 1) is independent from X. Deduce that (*) holds true. 2. We recall from the lecture notes that ety = ef [²0 ef #H(y) for all y € R. By applying this identity to y = Y, give a second proof of (*). 3. For s € R, compute E[e*X+tY] using the known expression for the Laplace transform of a Gaussian random variable, and then give a third proof of (*) by differentiating with respect to s.
Exercise I. Let X,Y~ N(0, 1) be jointly Gaussian random variables and let t€ R. The goal of this exercise is to prove the following identity, using three different ways: E[Xe] =tE[XY]E[e¹Y]. (*) 1. a. Prove that E[Xe¹x] = tE[e¹x]. b. Set p = E[XY]. Show that (X,Y) law (X, pX+√√1 - p²Z), where Z~ N(0, 1) is independent from X. Deduce that (*) holds true. 2. We recall from the lecture notes that ety = ef [²0 ef #H(y) for all y € R. By applying this identity to y = Y, give a second proof of (*). 3. For s € R, compute E[e*X+tY] using the known expression for the Laplace transform of a Gaussian random variable, and then give a third proof of (*) by differentiating with respect to s.
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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![Exercise I. Let X, Y ~ N(0, 1) be jointly Gaussian random variables and let t€ R. The
goal of this exercise is to prove the following identity, using three different ways:
E[Xe] =tE[XY] E[e¹]. (*)
1. a. Prove that E[XeX] = tE[e¹X].
b. Set p = E[XY]. Show that (X, Y) Law (X, pX+√√1 - p²Z), where Z~ N(0, 1)
is independent from X. Deduce that (*) holds true.
2. We recall from the lecture notes that ety=eH(y) for all y € R. By
applying this identity to y = Y, give a second proof of (*).
3. For s R, compute E[esX+tY] using the known expression for the Laplace transform
of a Gaussian random variable, and then give a third proof of (*) by differentiating
with respect to s.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7bf11458-9a82-4c45-a895-ac6b301730be%2F7680adb4-6735-4b28-866f-471d2db3f93d%2Fumizogk_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise I. Let X, Y ~ N(0, 1) be jointly Gaussian random variables and let t€ R. The
goal of this exercise is to prove the following identity, using three different ways:
E[Xe] =tE[XY] E[e¹]. (*)
1. a. Prove that E[XeX] = tE[e¹X].
b. Set p = E[XY]. Show that (X, Y) Law (X, pX+√√1 - p²Z), where Z~ N(0, 1)
is independent from X. Deduce that (*) holds true.
2. We recall from the lecture notes that ety=eH(y) for all y € R. By
applying this identity to y = Y, give a second proof of (*).
3. For s R, compute E[esX+tY] using the known expression for the Laplace transform
of a Gaussian random variable, and then give a third proof of (*) by differentiating
with respect to s.
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