(Solved):   pls provide correct solution with CLEAR HANDWRITING WITH FULL EXPLAINATION i will upvote ...

 

pls provide correct solution with CLEAR HANDWRITING WITH FULL EXPLAINATION i will upvote

Advanced Math Suppose \( f(x) \) is an even probability density so that \( f(x)=f(-x) \). Define \[ g(x)=\left\{\begin{array}{ll} \int_{x}^{\infty} \frac{f(s)}{s} d s, & \text { if } x>0 \\ g(-x), & \text { if } x<0 \end{array}\right. \] Why is \( \mathrm{g} \) a probability density? If has chf \( \Phi(t) \), how can you express the characteristic function of \( \mathrm{g} \) in terms of \( \Phi \) ?