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Suppose that we have n independent observations (X¡, Y¡) pf the random vector (X, Y) with pdf f(x, y). Show that (a) ƒ(y1....., Yn|X1, ..., Xn) = [I"-ı f(Vi|x;). This is called conditional independence. (b) ƒ(y¡|x1,..., Xn) = f(y¡|X¡) for i = 1,...,n. (c) f(yi, y¡\X1,..., Xn) = f(yi|x;¡)ƒ(y}\xj) for any i + j.

Suppose that we have n independent observations (X¡, Y¡) pf the random vector (X, Y) with pdf f(x, y). Show that (a) ƒ(y1....., Yn|X1, ..., Xn) = [I"-ı f(Vi|x;). This is called conditional independence. (b) ƒ(y¡|x1,..., Xn) = f(y¡|X¡) for i = 1,...,n. (c) f(yi, y¡\X1,..., Xn) = f(yi|x;¡)ƒ(y}\xj) for any i + j.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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A1 Please, show me how to compute this.
5. Suppose that we have n independent observations (X;, Y;) pf the random vector (X, Y) with pdf f(x, y). Show that (a) f(y1,..., Yn|x1,..., Xn) = II=1 f(Vi\xi). This is called conditional independence. (b) f(yi|x1,..., Xn) = f(yi|X¡) for i = 1,...,n. (c) f(yi, y¡\X1,..., Xn) = f(yi\xi)f (yj\x;) for any i + j. (d) If Var(Y1) <o then Cov(Y¡, Y¡|X1,.. , Xn) 1{0} (i – j)Var(Yı|X1).
Transcribed Image Text:5. Suppose that we have n independent observations (X;, Y;) pf the random vector (X, Y) with pdf f(x, y). Show that (a) f(y1,..., Yn|x1,..., Xn) = II=1 f(Vi\xi). This is called conditional independence. (b) f(yi|x1,..., Xn) = f(yi|X¡) for i = 1,...,n. (c) f(yi, y¡\X1,..., Xn) = f(yi\xi)f (yj\x;) for any i + j. (d) If Var(Y1) <o then Cov(Y¡, Y¡|X1,.. , Xn) 1{0} (i – j)Var(Yı|X1).
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