Part iii c) Let X and Y be two discrete random variables. We define Z = X + Y , i.e. Vw eΩ,Ζ(ω) = X(ω) + Y(ω) .i) Show that:P(Z = 2) = ) fx,x(x, z – x)ii) Now assume that X and Y are independent. Show that:P(Z = z) = ) fx(x)fr(z- x) = ) fx(z-y)fr(y)From now on, we assume that X and Y are independent random variables which havethe Poisson distributions with parameters Ax and Ay, respectively.iii) Show that Z has the Poisson distribution, with parameter Ax + Ay.

Given that, Let X and Y be two discrete random variables. Defined as Z=X+Y i.e.,…

Answered: c) Let X and Y be two discrete random…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 6AEXP

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Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.
c) Let X and Y be two discrete random variables. We define Z = X + Y, i.e. Vw e Ω,Ζ(ω) = Χ(ω) + Y(ω) . i) Show that: P(Z = 2) = ) _fx,r(x, z – x) ii) Now assume that X and Y are independent. Show that: P(Z = z) = ) fx(x)fy(z- x) = ) fx(z – y)fy(y) ニ From now on, we assume that X and Y are independent random variables which have the Poisson distributions with parameters Ax and Ay, respectively. iii) Show that Z has the Poisson distribution, with parameter Ax + Ay. iv) Show that the conditional distribution of X, given X+ Y = n , is binomial, and find its parameters.
Let X and Y be two discrete random variables. We define Z = X + Y, i.e. Vw E Q, Z(@) = X(@) + Y (@). i) Show that: P(Z = z) = ) fx.x(x, z – x) ii) Now assume that X and Y are independent. Show that: P(Z = z) = ) fx(x)fr(z- x) = )_fx(z-y)fr(y) X From now on, we assume that X and Y are independent random variables which have the Poisson distributions with parameters Ax and Ay, respectively. iii) Show that Z has the Poisson distribution, with parameter Ax + ly. iv) Show that the conditional distribution of X , given X+ Y = n , is binomial, and find its parameters.
4. Let X, Y be independent random variables. Prove that E(XY)= E(X) E(Y) and hence show that V(X+Y) = V(X)+V(Y).
= B1. Let X₁ and Y; be random variables with Var(X;) = o² and Var(Y₂) o for all ie {1,...,n}. Assume that each pair (X₁, Y;) has correlation Corr(X, Y) = p, but that (X₁, Y₂) and (X₁, Y;) are independent for all i j. (a) What is Cov(X₁, Yi) in terms of Ox, Oy (b) Show that Cov(X₁,Y) = poxoy/n, where Y is the average of the Y₁. (c) Determine Cov(X, Y). and p?
If X is a random variable, prove that Cov(X,X) = σX².
Suppose that X and Y are random variables with E(X) 2 , E(Y) = 5 and E(x?) = 8, E(Y?) = 30 and cov(2X – Y,4X – 3Y) = -12, then cov(-5X – 3,2Y + 4) is equal to: Hint: cov(ax + bY,cX + dY) = acV(X) + bdV(Y) + (bc + ad)cov(X,Y) O -120 O -59 O -64 O -30
Suppose that X and Y are random variables with E(X) = 2 , E(Y) = 5 and E(X²) = 8, E(Y?) = 30 and cov(X – 2Y,3X – 4Y) = -12, then cov(2X – 3, –5Y + 4) is equal to: Hint: cov(aX + bY,cx + dY) = acV(X) + bdV(Y) + (bc + ad)cov(X,Y) -59 -120 -64 -30
Q. For any random variables X and Y and the constant a, b, c and d show that Cov(aX + b, cY + d) = ac Cov(X, Y).

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