Mathematics

Problem

Consider the improper integral

I=\int_{0}^{\infty} x^{3}e^{-x}\,dx.

This type of integral appears when computing moments of continuous probability distributions. (a) Verify that the integral

I

converges. Provide a comparison argument using a familiar benchmark integral. (b) Evaluate

I

exactly. Show all steps leading to the closed-form numerical value. (c) Give the value of

I

to four decimal places and interpret your result in terms of the factorial or gamma function.