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.
$\therefore \, \, \, \int\limits_{-\infty}^{\infty} \, f(x) \, = \, 1$
Now we Have
$\int\limits_{-\infty}^{\infty} \, Ke^{-|x|} \, dx \, = \, 1 \, \Rightarrow \, \, 2 \int\limits_{0}^{\infty} \, K.e^{-|x|}dx = 1$
$\Rightarrow \, \, \, 2 \int\limits_{0}^{\infty} \, K . e^{-x} \, dx \, = \, 1$
$\Rightarrow \, \, -2 K . \bigg[e^{-x}\bigg]_{0}^{\infty} \, = 1 \, \Rightarrow \, 2K \, = \, 1$
$\Rightarrow \, \, K \, = \, \frac{1}{2}$