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 = 1a-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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