By Definition of Expectation of Random Variable:

$$ E(X)=\int_{-\infty}^{\infty}\,x\,f_X(x)\,dx $$

If $f_X(x)$ is Even, Then $E(X)=0$, Provided The Integral Exists, An Exception Being $\textbf{Cauchy Distribution}$ , Whose Mean Doesn't Exist.

Now the Question is, Will the PDF of Random Variable is Even ,When $E(X)=0$, I am Unable to Solve/Prove This. I Encourage all of you to Solve This...

$$ E(X)=\int_{-\infty}^{\infty}\,x\,f_X(x)\,dx $$

If $f_X(x)$ is Even, Then $E(X)=0$, Provided The Integral Exists, An Exception Being $\textbf{Cauchy Distribution}$ , Whose Mean Doesn't Exist.

Now the Question is, Will the PDF of Random Variable is Even ,When $E(X)=0$, I am Unable to Solve/Prove This. I Encourage all of you to Solve This...

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