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Let Xo, X₁, ... be i.i.d normal N(0, o²) random variables. Let the random variables Yo, Y₁, ... be defined as follows: Yo = Xo 3 Y₁ = =Y₁ Yo +=X₁ 4 Y/₂ = -5/ X₁ + 5 X ₂ 4 3 Yn = Yn-1 + x₂ : n = 1,2,3,... a. Show that uy (n) = 0 for n = 0,1,2,3 b. Find the pdfs fy (yo) fy₁ (₁), fy₂ (Y₂) c. Find the joint pdf, fy₁₂ (₁, 2)

Let Xo, X₁, ... be i.i.d normal N(0, o²) random variables. Let the random variables Yo, Y₁, ... be defined as follows: Yo = Xo 3 Y₁ = =Y₁ Yo +=X₁ 4 Y/₂ = -5/ X₁ + 5 X ₂ 4 3 Yn = Yn-1 + x₂ : n = 1,2,3,... a. Show that uy (n) = 0 for n = 0,1,2,3 b. Find the pdfs fy (yo) fy₁ (₁), fy₂ (Y₂) c. Find the joint pdf, fy₁₂ (₁, 2)

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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Please do part a-c and show work and steps
Let Xo, X₁, ... be i.i.d normal N(0, o2) random variables. Let the random variables Yo, Y₁, ... be defined as follows: Yo = Xo 4 Y₁ == Yo + = X₁ Y₂ : || 3 53 5 = 5+5X2 3 4 Yn = 5 Yn-1 + 55 Xn n = 1,2,3,... a. Show that uy (n) = 0 for n = 0,1,2,3 b. Find the pdfs fy (yo) fy₂ (₁), fy₂ (₂) c. Find the joint pdf, fy₁,₂ (₁, ₂)
Transcribed Image Text:Let Xo, X₁, ... be i.i.d normal N(0, o2) random variables. Let the random variables Yo, Y₁, ... be defined as follows: Yo = Xo 4 Y₁ == Yo + = X₁ Y₂ : || 3 53 5 = 5+5X2 3 4 Yn = 5 Yn-1 + 55 Xn n = 1,2,3,... a. Show that uy (n) = 0 for n = 0,1,2,3 b. Find the pdfs fy (yo) fy₂ (₁), fy₂ (₂) c. Find the joint pdf, fy₁,₂ (₁, ₂)
Hints: a. Use the equations for Yo, Y₁, Y2, Y3 to compute the means one by one. b. What do you know about the sum of independent Gaussian random variables? Recall that a Gaussian distribution is completely specified by its mean and variance. c. fy₁,₂ (₁,Y₂) = fy₂1Y, (y₂ly₁) fy, (v₁) C
Transcribed Image Text:Hints: a. Use the equations for Yo, Y₁, Y2, Y3 to compute the means one by one. b. What do you know about the sum of independent Gaussian random variables? Recall that a Gaussian distribution is completely specified by its mean and variance. c. fy₁,₂ (₁,Y₂) = fy₂1Y, (y₂ly₁) fy, (v₁) C
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