Q80 / 120 · VITEEE 2006 · Mathematics

The probability density $\mathrm{f}(\mathrm{x})$ of a continuous random variable is given by $f(x)=$ $\mathrm{Ke}^{-|\mathrm{x}|},-\infty < \mathrm{x} < \infty$. Then the value of $\mathrm{K}$ is

  • Correct answer
    A
    $\frac{1}{2}$
  • B
    2
  • C
    $\frac{1}{4}$
  • D
    4

Solution

Since $\mathrm{f}(\mathrm{x})$ is the probability density function of random variable $\mathrm{X}$. $\therefore \int_{-\infty}^{\infty} \mathrm{f}(\mathrm{x})=1$ Now we have $\begin{array}{l} \int_{-\infty}^{\infty} \mathrm{Ke}^{-|\mathrm{x}|} \mathrm{dx}=1 \Rightarrow 2 \int_{0}^{\infty} \mathrm{K} \cdot \mathrm{e}^{-|\mathrm{x}|} \mathrm{d} \mathrm{x}=1 \\ \Rightarrow 2 \int_{0}^{\infty} \mathrm{K} \cdot \mathrm{e}^{-\mathrm{x}} \mathrm{d} \mathrm{x}=1 \\ \Rightarrow-2 \mathrm{~K} \cdot\left[\mathrm{e}^{-\mathrm{x}}\right]_{0}^{\infty}=1 \Rightarrow 2 \mathrm{~K}=1 \\ \Rightarrow \mathrm{K}=\frac{1}{2} \end{array}$

Hence, the correct answer is option: 'A'.

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