Question

find the mean and standard deviation of a random variable with pdf f(x)=e^-x ,x>=0

Answer 1

mean=E(x)=$\int _0^{\infty \:} \:xe^{-x}dx=-e^{-x}x-e^{-x}$ |$\infty$ ₀=0-(-1)=1

E(x²)=$\int _0^{\infty \:}x^2e^{-x}dx$ =$-e^{-x}x^2+2\left(-e^{-x}x-e^{-x}\right)$ |∞ ₀=0-(-2)=2

Var=E(x²)-E(x)²=4-1=3

SD=$\sqrt {Var}$ = 1.73