If the probability density function of X is f(x)=1+ax2,−1≤x≤1,−1≤a≤1 then the expectation of X is? a) 6a b) a3 c) a2 d) 3a

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If the probability density function of X is f(x)=1+ax2,−1≤x≤1,−1≤a≤1 then the expectation of X is?  a) 6a  b) a3  c) a2  d) 3a

Javon Ross · Accepted Answer

Answer: E(X)=α3 Explanation: for a pdf f(x) the expectation of X E(X)=∫allxxf(x)dx E(X)=∫−11x(1+αx2)dx E(X)=12∫−11(x+αx2)dx E(X)=12[x22+αx33]−11 E(X)=12{[x22+αx33]1−[x22+αx33]0} E(X)=12{(⧸12+α3)−(⧸12−α3)} E(X)=12(α3+α3)=12×2α3 E(X)=α3

Hannah Escobar · Accepted Answer

Explanation:   By definition if f(x) is a continuous probability density function then:  E(X)=∫−∞∞xf(x)dx  So given that f(x)=1+ax2 for −1≤x≤1 then we have:  E(X)=∫−11x(1+ax2)dx  =12∫−11x(1+ax)dx  =12∫−11x+ax2dx  =12[x22+ax33]−11  =12{(12+a3)−(12−a3)}  =12(12+a3−12+a3)  =122a3  =a3

If the probability density function of X is f(x)=1+ax2,−1≤x≤1,−1≤a≤1 then the expectation of X is?...

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