Let X and Y be independent random variables, X~ Poisson (y) and Y~ Poisson(X); that is, for every pair of integers (i, j) such that 0 ≤ i and 0 ≤ j, P((X = i)n (Y = j)) = P(X = i) × P(Y = j). (a) Let S be the support of the random variable X + Y. Explicitly identify S. (b) For any k E S, find P(X + Y = k). (c) Fix k € S. For i≥ 0, find P(X= i| X +Y = k). (d) Fork E S, let g: S (-∞, +∞) be a function defined by g(k) = E(T), where for m > 0, T is the random variable such that P(T = m) = P(X = m X + Y = k). Derive a simple algebraic expression for g(k). (e) Find E[g(X+Y)]. (f) Find Var[g(X+Y)]. (g) For k = S, let h: S (-∞, +∞) be a function defined by h(k) = Var(T). Derive a simple algebraic expression for h(k). (h) Find E[h (X+Y)]. (i) Obtain a simple expression for E[h(X+Y)] + Var[g(X+Y)].
Let X and Y be independent random variables, X~ Poisson (y) and Y~ Poisson(X); that is, for every pair of integers (i, j) such that 0 ≤ i and 0 ≤ j, P((X = i)n (Y = j)) = P(X = i) × P(Y = j). (a) Let S be the support of the random variable X + Y. Explicitly identify S. (b) For any k E S, find P(X + Y = k). (c) Fix k € S. For i≥ 0, find P(X= i| X +Y = k). (d) Fork E S, let g: S (-∞, +∞) be a function defined by g(k) = E(T), where for m > 0, T is the random variable such that P(T = m) = P(X = m X + Y = k). Derive a simple algebraic expression for g(k). (e) Find E[g(X+Y)]. (f) Find Var[g(X+Y)]. (g) For k = S, let h: S (-∞, +∞) be a function defined by h(k) = Var(T). Derive a simple algebraic expression for h(k). (h) Find E[h (X+Y)]. (i) Obtain a simple expression for E[h(X+Y)] + Var[g(X+Y)].
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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Please answer a-i, as all parts are related.
![Let X and Y be independent random variables, X~ Poisson (y) and Y~
Poisson(y) and Y~ Poisson (X);
that is, for every pair of integers (i, j) such that 0 ≤ i and 0 ≤ j,
P(( X = i) n (Y = j)) = P( X = i) × P(Y = j).
(a) Let S be the support of the random variable X + Y. Explicitly identify S.
(b) For any k € S, find P(X + Y = k).
(c) Fix k = S. For i ≥ 0, find P(X= i| X + Y = k).
(d) For k = S, let g : S → ( − ∞, +∞) be a function defined by g(k) = E(Tk), where
for m≥ 0, T is the random variable such that P(T = m) = P(X= m X + Y = k).
Derive a simple algebraic expression for g(k).
(e) Find E[g(X + Y)].
(f) Find Var[g(X + Y)].
(g) For k = S, let h : S → ( − ∞, +∞) be a function defined by h(k) = Var(Tk).
Derive a simple algebraic expression for h(k).
(h) Find E[h (X+Y)].
(i) Obtain a simple expression for E[h(X + Y)] + Var[g(X+Y)].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b1ffeba-917b-4621-8d66-7fb659f36c04%2F6c05d5b3-bfbd-4e4e-9b72-54286fb63417%2Fo7uq54p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let X and Y be independent random variables, X~ Poisson (y) and Y~
Poisson(y) and Y~ Poisson (X);
that is, for every pair of integers (i, j) such that 0 ≤ i and 0 ≤ j,
P(( X = i) n (Y = j)) = P( X = i) × P(Y = j).
(a) Let S be the support of the random variable X + Y. Explicitly identify S.
(b) For any k € S, find P(X + Y = k).
(c) Fix k = S. For i ≥ 0, find P(X= i| X + Y = k).
(d) For k = S, let g : S → ( − ∞, +∞) be a function defined by g(k) = E(Tk), where
for m≥ 0, T is the random variable such that P(T = m) = P(X= m X + Y = k).
Derive a simple algebraic expression for g(k).
(e) Find E[g(X + Y)].
(f) Find Var[g(X + Y)].
(g) For k = S, let h : S → ( − ∞, +∞) be a function defined by h(k) = Var(Tk).
Derive a simple algebraic expression for h(k).
(h) Find E[h (X+Y)].
(i) Obtain a simple expression for E[h(X + Y)] + Var[g(X+Y)].
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