Question

Estimating the intensity of a radiation source. 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:

p Y∣X (y∣x)= {e^-x/y, if y = 0,1,2...., otherwise

The parameter x is unknown and is modeled as the value of a random variable X that is exponentially distributed with parameter μ>0 :
f X(x)= {μe^-μx, if x ≥ 0, otherwhise

(a) Find the MAP estimate of X based on the observed value y of Y. Express your answer in terms of y and μ.

Answer 1

The student is asking for the MAP estimate for a parameter of a Poisson distribution, given that the parameter itself has an exponential distribution.

Step-by-step explanation:

The student's question involves calculating the maximum a posteriori (MAP) estimate for a parameter of a Poisson distribution, given the parameter is itself a random variable with an exponential distribution. The Poisson distribution is used to model the number of events in fixed intervals of time or space when these events occur with a known average rate and independently of each other. The MAP estimation combines the prior distribution of the parameter (exponential in this case), with the likelihood of the observed data to estimate the most probable value of the parameter after observing the data.

To calculate the MAP estimate for the parameter X based on the observed value y of Y, we need to find the mode of the posterior distribution. The posterior distribution is obtained by applying Bayes' theorem, which in this case involves multiplying the Poisson likelihood function by the exponential prior and then normalizing.

However, a full solution to compute the MAP estimate requires additional steps involving calculus, which were not provided in the student's question and hence cannot be fully derived here without further information.