Question

A probability density function is of the form $p\left(x\right)=K{e}^{-\alpha \left|x\right|},xϵ(-\infty ,\infty ),$ the value of K is

• A. 0.5
• B. 1
• C. $0.5\alpha$
• D. $\alpha$

Answer 1

The correct option is C $0.5\alpha$

$p\left(x\right)=k{e}^{-\alpha |x|},ϵ(-\infty ,\infty )$

Since p(x) is a pd.f.

$\therefore {\int }_{-\infty }^{\infty }p\left(x\right)=1$

$\Rightarrow {\int }_{-\infty }^{\infty }k{e}^{-\alpha |x|}dx=1$

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

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

$\frac{2K}{-\alpha }{\left[e^{-\alpha x}\right]}_0^{\infty }=1$

$\frac{2K}{\alpha }=1$

$K=\left(0.5\right)\alpha$