Probability | Part IA, 2006

Let γ>0\gamma>0 and define

f(x)=γ11+x2,<x<f(x)=\gamma \frac{1}{1+x^{2}}, \quad-\infty<x<\infty

Find γ\gamma such that ff is a probability density function. Let {Xi:i1}\left\{X_{i}: i \geqslant 1\right\} be a sequence of independent, identically distributed random variables, each having ff with the correct choice of γ\gamma as probability density. Compute the probability density function of X1++X_{1}+\cdots+ XnX_{n}. [You may use the identity

m{(1+y2)[m2+(xy)2]}1dy=π(m+1){(m+1)2+x2}1m \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 xRx \in \mathbb{R} and mNm \in \mathbb{N}.]

Deduce the probability density function of


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

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