A particular species of mollusk is distributed according to a Poisson process with an unknown density of lambda per cubic meter of water. A sensor is constructed that can detect these species that are within 4 m, and n readings are collected over a year to try and measure lambda. These readings are independent, and so the number of readings that detect a mollusk has the distribution X~Bi(n,p) for some unknown value of p.
p is estimated to be approximately 0.7.
Problem: Let Y be the distance from the sensor to the nearest mollusk.
Show that the c.d.f. of Y is of the form
F(y) = 1- e-(lambda)(4/3pi y3 if y >0.
b) Find the p.d.f. of Y and use it to compute E(Y).
gamma(4/3) = 0.8929795
The Attempt at a Solution
I'm not so much worried about part a), but I tried part b) and I'm getting nowhere. I took the derivative of F(y) to get f(y) (the p.d.f) and got f(y) = 4pi(lambda) y^2(e-(lambda)(4/3 pi*y3, and then for E(Y), I multiplied that by y and took the integral from 0 to infinity of that. I read in my textbook that for the basic proof of expected value, you're supposed to make a substitution for theta = 1/lambda, and then substitute more values to make this integral look like the gamma function. So I substituted x = 4pi/3theta y^3 (to make the power of e = -x) and then what im left with after simplifying is... the integral from 0 to infinity of e^-x dx... I'm pretty sure thats wrong though... since the problem tells me I should be using the gamma function of 4/3 for something.. Can anyone help me with this? Thanks.