Question

$P_x\left(X\right)=M{e}^{(-2|x|)}+N{e}^{(-3|x|)}$ is the probability density function for the real random variable X, over the entire x-axis, M and N are both positive real numbers. The equation relating M and N is

• A. $M+\frac{2}{3}N=1$
• B. $2M+\frac{1}{3}N=1$
• C. M + N = 1
• D. M + N = 3

Answer 1

The correct option is A $M+\frac{2}{3}N=1$

$\because P_x\left(X\right)\text{is p.d.f}$

$\Rightarrow x{\int }_{-\infty }^{\infty }{P}_x\left(x\right)dx=1$

${\int }_{-\infty }^{\infty }[M{e}^{-2|x|}+N{e}^{-3|x|}]dx=1$

$\Rightarrow 2{\int }_0^{\infty }[M{e}^{-2|x|}+N{e}^{-3|x|}]dx=1$
$(\because \text{integrand is even})$

$\Rightarrow 2M{\int }_0^{\infty }{e}^{-2x}dx+2N{\int }_0^{\infty }{e}^{-3x}dx=1$

$(\because |x|=x\text{for}x>0)$

$\Rightarrow M+2\frac{N}{3}=1$