Question

1. Let X be a random variable with probability density function fX given by fX(x) = γαγ/ (x + α)^γ+1 , x ≥ 0,

0, x < 0,

where α > 0 and γ > 0.

(a) Find the cumulative distribution function (cdf) FX of X.

(b) Let Y = log(X+α /α) . Find the cdf of Y and identify the distribution.

(c) How could a realisation of X be generated from an R(0,1) random number generator?

(d) Let Z = min(X,M), where M > 0 is a fixed constant. Derive the cdf FZ of Z and compute its mean.

Answer 1

We are given a random variable X with pdf :

(a)

The cdf of X is given by:

(b)

The cdf of Y = log((X+a)/a) is given by:

Thus, Y has a Weibull distribution with shape parameter 2 and scale parameter 1

(c)

A realisation of X can be generated by Inverse transform method, we equate the cdf of X equal to U (where U is R(0,1)). Then we solve for X.

The variables generated from the above equation will have the required distribution.

(d)

CDF of Z:

Now,

where

Thus,

Mean of Z:

Now,

Now, the expression inside the integral is same as f_X(x) with parameter y replaced by (y-1) thus if we assume y>1, then the integral will have the same form as cdf of X with paramter y replaced by (y-1). Thus, we get:

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