GATE (TF) Textile 2017 Question Paper Solution | GATE/2017/TF/26

Question 26 (Textile Engineering & Fibre Science)

The median, accurate to two decimal places of a random variable $X$ with the probability density $f(x) = \begin{cases} 3x^{-4} & x>1 \\ 0 & x\leq 1 \end{cases}$ ; is __1.25 to 1.27__.

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For median-
PDF= $\frac{1}{2}$
$\int_{-\infty}^{m} f(x) dx=\frac{1}{2}$
$\int_{-\infty}^{m} (3x^{-4}) dx=\frac{1}{2}$
$3[\frac{x^{-4+1}}{-4+1}]_{-\infty}^{m}=\frac{1}{2}$
$3[(\frac{x^{-3}}{-3})]_{-\infty}^{m}=\frac{1}{2}$
$[(x^{-3})]_{-\infty}^{m}=\frac{1}{2}$
$[0-m^{-3}]=\frac{1}{2}$
$[m^{-3}]=\frac{1}{2}$
$m^3=2$
$m=1.26$ (Ans)

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