1) Let $\displaystyle \Phi (x)$ a real valued function with continuous and positive second derivative in every point.

Let $\displaystyle X$ be a random variable such that its expectation and that of the random variable $\displaystyle \Phi (X)$ exists.

Prove that $\displaystyle \Phi (EX) \leq E\Phi (X)$

2) Prove that if $\displaystyle X_n$ converges to $\displaystyle X$ in $\displaystyle L^p$ norm then $\displaystyle X_n$ converges to $\displaystyle X$ in probability.

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