Question

Let h be an exponentially-distributed random variable with the distribution function p- exp(-x) for x > 0 and ph = nction Ph 0 for a s 0. Derive the distribution function of its square root, Solution: 2y exp(-y2

Answer 1

here let Y=√x

therfore cdf of Y =F(Y)=P(Y<y)=P(√X <y)=P(X<y²)=$\int_{0}^{y^{2}}$f(x) dx =$\int_{0}^{y^{2}}$exp(-x) dx =-exp(-x)|^{y^2}₀

F(y)=1-exp(-y²)

therefore distribution function of y: f(y)=(d/dy)*F(y)=(d/dy)*(1-exp(-y²))

f(y)=2yexp(-y²)