JAM 2022 [ 51-60 ]

Let X be a random variable having the probability density function

f(x) = \left\{ \begin{matrix} ax^2 +b & , 0 \leq x \leq 3 \\ 0 &, otherwise \end{matrix} \right. \\ \text{where $a$ and $b$ are real constants, and } P(X\geq2) = \frac{2}{3}.

Then E(X) equals _____________________ (round off to 2 decimal places)

\int_{x=0}^{3} (ax^2+b) dx = 1 \\ \implies 9a + 3b =1 \\ Pr(X \geq 2) = \frac{2}{3} \\ Pr(X < 2) = \frac{1}{3} \\ \implies \frac{8a}{3} + 2b = \frac{1}{3} \\ \implies 8a + 6b = 1

a-3b= 0 \implies a=3b \\ 30b=1 \\ b= \frac{1}{30} , a=\frac{1}{10}

E(X) = \int_{x=0}^{3} x(ax^2+b) dx = \frac{81a}{4} + \frac{9b}{2} = \frac{81}{40} + \frac{6}{40} = \frac{87}{40}= 2.175

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