Math Problem Statement
what if f(y) = exp(y(theta) - e^(theta) - ln(y!)), find E[ln(y)] in terms of theta
Solution
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Probability Theory
Poisson Distribution
Expected Value
Special Functions
Digamma Function
Formulas
f(y) = exp(yθ - e^θ - ln(y!))
Poisson PMF: P(Y = y) = (e^(-λ) * λ^y) / y!, where λ = e^θ
E[ln(Y)] = ψ(λ), where ψ is the digamma function
Theorems
Poisson Distribution Theorem
Digamma Function Definition
Suitable Grade Level
Advanced Undergraduate/Graduate Level
Related Recommendation
Expected Value and Variance of Transformed Random Variable S = ln(R)
Understanding the Law of Large Numbers and Expected Value in Probability Theory
Law of Large Numbers and Expected Value in Probability Theory
Accuracy of E[ln(Y^θ)] ≈ θ*E[ln(Y)] Approximation for Small θ
Calculating the Expected Value for Insurance Losses Using Gaussian PDF