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Q)

Find the mean and variance for the following probability density functions $f(x) = \left\{ \begin{array}{l l} xe^{-x}, & \quad \text{if $x$$>$0}\\ 0, & \quad \text{otherwise} \end{array} \right.$

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A)
Toolbox:
  • Let X be a continuous random variable with probability density function f(x). Then the mathematical expectation of X is defined as $E(X)=\int_{-\infty}^\infty x f(x)dx$
  • $E(\phi (X))=\int_{\infty}^{\infty}\phi(x) f(x)dx$
  • Var$(X)=E(X^2)-[E(X)]^2$
  • $E(c)=c$
  • $E(aX\pm b)=aE (X)\pm b$
Step 1:
$E(X)=\int_{-\infty}^\infty x f(x) dx$
$\qquad=\int_0^\infty x^2 e^{- x}dx$
$\qquad=2!$
$\qquad=2$(Gamma integral)
Step 2:
$E(X^2)=\int_{\infty}^\infty x^2f(x)dx$
$\qquad=\int_0^{\infty} x^3e^{-x} dx$
$\qquad=3!$
$\qquad=6$(Gamma integral)
Step 3:
Var$(X)=E(X^2)-[E(X)]^2$
$\qquad\;\;\;=6-4$
$\qquad\;\;\;=2$
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