Question
Download Solution PDFIf a random communication signal has PDF given by P(X) = 2ae-b|x| for all x, -∞ < x < +∞ then :
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
For any valid probability density function (PDF), the total area under the curve must be 1:
\( \int_{-\infty}^{\infty} P(x)\, dx = 1 \)
Given:
\(P(x) = 2ae^{-b|x|} \quad \text{for all } x \in (-\infty, \infty)\)
Calculation:
Since the function is even (depends on |x|), integrate over the positive side and double it:\[ \int_{-\infty}^{\infty} 2ae^{-b|x|} dx = 2 \cdot \int_0^{\infty} 2ae^{-bx} dx = 4a \cdot \int_0^{\infty} e^{-bx} dx \]
\(= 4a \cdot \left[ \frac{1}{b} \right] = \frac{4a}{b}\)
Set equal to 1 (normalization condition):
\[ \frac{4a}{b} = 1 \Rightarrow b = 4a \]
Hence, the correct option is 2
Last updated on Jun 24, 2025
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