Question

Given that x is a random variable in range $[0,\infty ]$ with a probability density function $e^{-x/2}/K,$ the value of the constant K is ______ .

• A. 2

Answer 1

The correct option is A 2

For p.d.f, ${\int }_{-\infty }^{\infty }f\left(x\right)dx=1$

$\Rightarrow {\int }_0^{\infty }\frac{e^{\frac{-x}{2}}}{K}dx=1$

$\Rightarrow \frac{1}{K}{\left(\frac{e^{\frac{-x}{2}}}{(-\frac{1}{2})}\right)}_0^{\infty }=1$

$\Rightarrow -\frac{2}{K}{\left(e^{-\frac{x}{2}}\right)}_0^{\infty }=1$

$\Rightarrow -\frac{2}{K}(0-1)=1$
$\Rightarrow K=2$