Question

The probability density f(x) of a continuous random variable is given by f(x) = Ke${}^{|x|}$, $-\infty <x<\infty$Then the value of K is

• A. $\frac{1}{2}$
• B. 2
• C. $\frac{1}{4}$
• D. 4

Answer 1

Answer: (A)

Since f(x) is the probability density function of random variable X.
$\therefore \underset{-\infty }{\overset{\infty }{\int }}f(x)=1$
Now we Have
$\underset{-\infty }{\overset{\infty }{\int }}K{e}^{-|x|}dx=1\Rightarrow 2\underset{0}{\overset{\infty }{\int }}K.e^{-|x|}dx=1$
$\Rightarrow 2\underset{0}{\overset{\infty }{\int }}K.e^{-x}dx=1$
$\Rightarrow -2K.[e^{-x}{]}_0^{\infty }=1\Rightarrow 2K=1$
$\Rightarrow K=\frac{1}{2}$