Math Problem Statement
Suppose X is an exponential random variable with density function fX(x) = e−x, x > 0. 1 Also, the conditional density of Y given X is Poisson with mean X; i.e. xn −x P(Y =n|X =x)= n! e , n∈{0,1,2,...}. Find P(Y = 1). Also, find the generating function E tY and hence find the expectation of Y , confirming your answer with a direct calculation of the mean viaE(Y)=E[E(Y |X)].
Solution
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Random Variables
Poisson Distribution
Exponential Distribution
Generating Functions
Law of Total Probability
Formulas
P(Y = 1) = ∫ P(Y = 1 | X = x) * fX(x) dx
P(Y = 1 | X = x) = x * e^(-x)
E[t^Y] = ∫ e^(x(t - 1)) * fX(x) dx
E[Y] = E[E(Y | X)]
Theorems
Law of Total Expectation
Moment Generating Function
Integration by Parts
Suitable Grade Level
Undergraduate (Advanced Probability/Statistics)
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