Let the probability density function of a random variable x be given as

f(x) = ae-2|x|

The value of ‘a’ is __________.

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GATE EE 2022 Official Paper

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Answer 1

Concept:

The probability density function for any f(x) is defined as:

$\int_{- \infty}^{+ \infty} f (x) d x = 1$

Modulus function is f(x) = |x| is defined as:

$f (x) = {\begin{matrix} - x; & - \infty < x < 0 \\ + x; & 0 < x < \infty \end{matrix}}$

Calculation:

$\int_{- \infty}^{+ \infty} f (x) d x = 1$

$\int_{- \infty}^{0} a e^{- 2 (- x)} d x + \int_{0}^{\infty} a e^{- 2 (+ x)} d x = 1$

$\frac{a}{2} (e^{2 x})_{- \infty}^{0} + \frac{a}{2} (- e^{- 2 x})_{0}^{\infty} = 1$

$\frac{a}{2} {e^{0} - e^{- \infty}} + \frac{a}{2} {- e^{- \infty} - (- e^{0})} = 1$

$\frac{a}{2}$ (1-0) + $\frac{a}{2}$ (0+1) = 1

a = 1

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