GATE (TF) Textile 2014 Question Paper Solution | GATE/2014/TF/29

Question 29 (Textile Engineering & Fibre Science)

$X$ is a continuous random variable whose probability density function is given by

$f(x)=\begin{cases}K(4x-2x^2) & for \:\:\:0<x<2 \\ 0 & otherwise \end{cases}$

The value of the constant $K$ , accurate to three decimal places, is……

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$f(x)=\begin{cases}K(4x-2x^2) & for \:\:\:0<x<2 \\ 0 & otherwise \end{cases}$

$\int_{0}^{2} K(4x-2x^2) dx=1$
$4K\int_{0}^{2} x dx-2K\int_{0}^{2} x^2 dx=1$
$4K[\frac{x^2}{2}]_{0}^{2}-2K[\frac{x^3}{3}]_{0}^{2}=1$
$4K[\frac{2^2}{2}-0]-2K[\frac{2^3}{3}-0]=1$
$4K[2]-2K[\frac{8}{3}]=1$
$8K-K[\frac{16}{3}]=1$
$\frac{8}{3} K=1$
$K=\frac{3}{8}$
K=0.375 (Ans)

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