Measure, Integration & Real Analysis · Problem 12.15

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · density-function-computation

Problem

Suppose

\alpha > 0

and

h(x) = \begin{cases} 0 & \text{if } x < 0, \\ \alpha^2 x e^{-\alpha x} & \text{if } x \geq 0. \end{cases}

Let

P = h\,d\lambda

and let

X

be the random variable defined by

X(x) = x

for

x \in \mathbf{R}

. (a) Verify that

\int_{-\infty}^{\infty} h\,d\lambda = 1

. (b) Find a formula for the distribution function

\widetilde{X}

. (c) Find a formula (in terms of

\alpha

) for

EX

. (d) Find a formula (in terms of

\alpha

) for

\sigma(X)