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A probability density function density function is of form P(x)= k e ^(-a |x|) , the value of k is

A)0.5

B)1

C) 0.5 a

D) a

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Ans C) 0.5a (assuming X $\epsilon$ (-$\infty$, $\infty$))

P(X) = ke(-a|x|)

$\therefore \int_{-\infty }^{\infty}$ ke(-a|x|) dx = 1

$\Rightarrow$ 2$\int_{0}^{\infty }$ ke(-a|x|) dx = 1 ($\because$ ke(-a|x|) is even function)

$\Rightarrow$ 2k $\int_{0}^{\infty }$ e-ax dx = 1

$\Rightarrow$ 2k  [ $\frac{e^{-ax}}{-a}$ ]0$\infty$ = 1

$\Rightarrow$ $\frac{2k}{-a}$ * (0 - 1) = 1

$\Rightarrow$ k = 0.5a

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