Linear Minimum Mean Square Estimator

[sant142]

I am not able to understand how to go about this problem:

Find the minimum mean square estimator for the scalar parameter w based
on the scalar observation z = ln w + n where

f(w) =1 if 0<=w<=1;
0 else:

and
f(n) =e^-n if n>= 0;
0 else

I did f(z/w) = (f(n))/ g'(n) at n = z- ln w

Am i wrong?

How to go about it?

 

[Valkarie]

The minimum mean square estimator for a scalar parameter w based on the scalar observation z = ln w + n is given by the following equation:E[w|z] = e^(z-n)This equation is derived from the conditional probability density of w given z, which can be written as:f(w|z) = f(z-ln w)f(w)/g'(z-ln w)where f(z-ln w) is the probability density function of n and g'(z-ln w) is the derivative of the cumulative distribution function of n. The minimum mean square estimator is then derived by taking the expected value of w given z:E[w|z] = ∫wf(w|z)dw = ∫we^(z-n)f(w)dw/g'(z-ln w)The integral can be evaluated using the substitution u = ln w, yielding the result:E[w|z] = e^(z-n)/g'(z-ln w)