Answer to Question #316479 in Statistics and Probability for Sohan

Question

Find the expected value of the random variable X and its variance having the following density function:

f(x) = 5 (1-x4) 0<x<2

= 0 elsewhere

Full details math and explain.

Answer 1

Expected value

$E(X)=5∫_0^2x(1-x^4 )dx=5∫_0^2x-x^5 dx=5[\frac{x^2}{2}-\frac{x^6}{6}]_0^2=5[(\frac{2^2}{2}-\frac{2^6}{6})-0]=-\frac{130}{3}$

Variance

$Var(X)=E(X^2)-(E(X))^2$

$E(X^2)=5∫_0^2x^2 (1-x^4 )dx=5∫_0^2x^2-x^6 dx=5[\frac{x^3}{3}-\frac{x^7}{7}]_0^2=5[(\frac{2^3}{3}-\frac{2^7}{7})-0]=-\frac{1640}{21}$

$Var(X)=-\frac{1640}{21}-(-\frac{130}{3})^2=-\frac{123220}{63}=-1955.8730$