Question involving pdf of change of variable

[alex07966]

A Geiger counter records the times at which radioactive particles are detected. However, if two particles occur within a short time θ of each other, the second one is not detected. Then, the probability density function of X, the time between successive recorded detections, is f(x) = k exp(−x/μ), for θ ≤ x < ∞.

Find k and the expectation of X.

Let Y be the smallest of a sample of n observations on X. Find the probability density function of Y .

I managed the first part and got k = (1/μ)exp(θ/μ) and E(X) = θ + μ

I'm not exactly sure how to begin the second part!? I know there are n x_i's in the range of X and that Y is the smallest of these x_i's, but then how do it write that? confused :s

Any help would be great.

 

[lippyka]

For the second part, let Y be the smallest of the sample of n observations on X. The probability density function of Y is given by:fY (y) = n * fX (y) * FX (y)^(n−1) where FX (y) is the cumulative density function of X. Therefore, the probability density function of Y is fY (y) = n * k * exp(-y/μ) * (1 - exp(-y/μ))^(n−1) for θ ≤ y < ∞.