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$f(x) = \begin{Bmatrix} cx & if (0<x<4) \\ 0 & otherwise \end{Bmatrix}$

(c) Determine EX and Var(X).
in Probability by Boss (10.9k points) | 40 views

1 Answer

+1 vote
$E[X] = \Large  \int_{0}^{4} x \ cx$

$E[X] = \Large  \int_{0}^{4} \frac{x^2}{8}$

$E[X] = \Large \frac{1}{8} \int_{0}^{4} x^2$

$E[X] = \Large \frac{1}{8} [\ \frac{x^3}{3} ]_{0}^{4}$

$E[X] = \Large \frac{1}{8} [\ \frac{4^3}{3} ]$

$E[X] = \Large  \frac{8}{3} $

and $E[X^2] = \Large  \int_{0}^{4} x^2 \ cx$

$E[X^2] = \Large\frac{1}{8}  \int_{0}^{4} x^3$

$E[X^2] = \Large\frac{1}{8}  [ \frac{x^4}{4}]_0^4$

$E[X^2] = \Large\frac{1}{8}  [ \frac{4^4}{4}] = 8$

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

$Var[X] = 8 - (\frac{8}{3})^2$

$Var[X] = \Large  \frac{8}{9}$
by Boss (36.7k points)
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