Question

Suppose Y is a random variable with pdf f(y)=ay²-b,0<y<3

Find a and b given that E(Y)=15/7

Answer 1

$$\displaystyle\int_{0}^3(ay^2-b)=1$$

$$[\dfrac{ay^3}{3}-by]\begin{matrix} 3 \\ 0 \end{matrix}=1$$

$$9a-3b=1$$

$$E(Y)=\displaystyle\int_{0}^3y(ay^2-b)=[\dfrac{ay^4}{4}-\dfrac{by^2}{2}]\begin{matrix} 3 \\ 0 \end{matrix}$$

$$=\dfrac{81a}{4}-\dfrac{9b}{2}=\dfrac{15}{7}$$

$$b=3a-\dfrac{1}{3}$$

$$27a-18a+2=\dfrac{20}{7}$$

$$9a=\dfrac{6}{7}$$

$$a=\dfrac{2}{21}$$

$$b=-\dfrac{1}{21}$$