Let $\gamma>0$ and define

$$f(x)=\gamma \frac{1}{1+x^{2}}, \quad-\infty<x<\infty$$

Find $\gamma$ such that $f$ is a probability density function. Let $\left\{X_{i}: i \geqslant 1\right\}$ be a sequence of independent, identically distributed random variables, each having $f$ with the correct choice of $\gamma$ as probability density. Compute the probability density function of $X_{1}+\cdots+$ $X_{n}$. [You may use the identity

$$m \int_{-\infty}^{\infty}\left\{\left(1+y^{2}\right)\left[m^{2}+(x-y)^{2}\right]\right\}^{-1} d y=\pi(m+1)\left\{(m+1)^{2}+x^{2}\right\}^{-1}$$

valid for all $x \in \mathbb{R}$ and $m \in \mathbb{N}$.]

Deduce the probability density function of

$$\frac{X_{1}+\cdots+X_{n}}{n}$$

Explain why your result does not contradict the weak law of large numbers.