Calculating Risk Function of Normal Distribution Sample

[kimkibun]

Homework Statement

Let X₁,X₂ be a random sample of size 2, from a normal distribution with mean μ and σ²=1. Let T=t(X₁,X₂)=X₁+X₂ be the estimator of μ and l(t;μ)=t-μ be the loss function. Compute risk function R(μ).

Homework Equations

R(μ)=∫∫(l(t;μ))∏f(x_i;μ)dx_i

The Attempt at a Solution

Since we have a normal distribution, then

∏f(x_i;μ)=(1/2∏)exp(-1/2)[(x₁-μ)²+(x₂-μ)²]

l(t;μ)=t-μ=(X₁+X₂)-μ

so the risk function is given by,

R(μ)=∫∫[(X₁+X₂)-μ](1/2∏)exp(-1/2)[(x₁-μ)²+(x₂-μ)²]dx₁dx₂

i just want to know if my equation for risk function is correct.

 

[nate808]

Hi there! Your equation for the risk function looks correct to me. However, I would suggest double-checking your calculations to make sure everything is accurate. Also, you may want to consider simplifying the expression further to make it more manageable. Good luck!