GATE (TF) Textile 2010 Question Paper Solution | GATE/2010/TF/44

Question 44 (Textile Engineering & Fibre Science)

If the density function of a random variable X is given by

    \[f(x)=\begin{cases}\frac{x}{2} & 0<x<2 \\ 0 & otherwise\end{cases}\]

then the value of X will be

(A)4/3
(B)1/2
(C)0
(D)1/6
[Show Answer]

    \[f(x)=\begin{cases}\frac{x}{2} & 0<x<2 \\ 0 & otherwise\end{cases}\]

Mean=\int_{0}^{2} xf(x) dx

Mean=\int_{0}^{2} x \times \frac{x}{2} dx

Mean=\int_{0}^{2} \frac{x^2}{2} dx

Mean=\frac{1}{2} \times \int_{0}^{2} x^2 dx

Mean=\frac{1}{2} \times [\frac{x^3}{3}]_{0}^{2}

Mean=\frac{1}{6} \times [x^3]_{0}^{2}

Mean=\frac{1}{6} \times [2^3-0]

Mean=\frac{1}{6} \times 8

Mean=\frac{8}{6}

Mean=\frac{4}{3} (Ans)

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