Topic: Random Processes
If a random communication signal has PDF given by $P(X)=2ae^{-b|x|}$ for all x, $-\infty < x < +\infty$ then:
- A. $b=2a$
- B. $b=4a$
- C. $b=a/2$
- D. $b=0$
Answer: B. $b=4a$
Explanation: The total area under a Probability Density Function (PDF) curve must be equal to 1.
$\int_{-\infty}^{\infty} 2ae^{-b|x|} dx = 1$
Since the function is even, $2 \int_{0}^{\infty} 2ae^{-bx} dx = 1$
$4a \left[\frac{e^{-bx}}{-b}\right]_{0}^{\infty} = 1 \Rightarrow 4a(0 - (\frac{-1}{b})) = 1 \Rightarrow \frac{4a}{b} = 1 \Rightarrow b = 4a$.