Question #185739

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


1
Expert's answer
2021-05-07T09:29:06-0400
03(ay2b)=1\displaystyle\int_{0}^3(ay^2-b)=1

[ay33by]30=1[\dfrac{ay^3}{3}-by]\begin{matrix} 3 \\ 0 \end{matrix}=1

9a3b=19a-3b=1

E(Y)=03y(ay2b)=[ay44by22]30E(Y)=\displaystyle\int_{0}^3y(ay^2-b)=[\dfrac{ay^4}{4}-\dfrac{by^2}{2}]\begin{matrix} 3 \\ 0 \end{matrix}

=81a49b2=157=\dfrac{81a}{4}-\dfrac{9b}{2}=\dfrac{15}{7}

b=3a13b=3a-\dfrac{1}{3}

27a18a+2=20727a-18a+2=\dfrac{20}{7}

9a=679a=\dfrac{6}{7}

a=221a=\dfrac{2}{21}


b=121b=-\dfrac{1}{21}



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