Caleb builds a particle detector and uses it to measure radiation from a remote star. On any given day, the number of particles, Y, that hit the detector is distributed according to a Poisson distribution with parameter x. The parameter x is unknown and is modeled as the value of a random variable X that is exponentially distributed with parameter μ>0:

fX(x)={μe−μx,0,if x≥0,otherwise.
The conditional PMF of the number of particles hitting the detector is

pY∣X(y∣x)=⎧⎩⎨e−xxyy!,0,if y=0,1,2,…,otherwise.
(a) Find the MAP estimate of X based on the observed value y of Y. Express your answer in terms of y and μ. Use 'mu' to denote μ.

x^MAP(y)=- unanswered
(b) Our goal is to find the LMS estimate for X based on the observed particle count y.

We can show that the conditional PDF of X given Y is of the form

fX∣Y(x∣y)=λy+1y!xye−λx,x>0,y≥0.
Express λ as a function of μ. You may find the following equality useful:

∫∞0ay+1xye−axdx=y!,for any a>0.
λ=- unanswered
Find the LMS estimate of X based on the observed particle count y. Express your answer in terms of y and μ. Hint: You may want to express xfX∣Y(x∣y) in terms of fX∣Y(x∣y+1).

x^LMS(y)= - unanswered