Paper 2, Section I, F

(a) State the Cauchy-Schwarz inequality and Markov's inequality. State and prove Jensen's inequality.

(b) For a discrete random variable $X$, show that $\operatorname{Var}(X)=0$ implies that $X$ is constant, i.e. there is $x \in \mathbb{R}$ such that $\mathbb{P}(X=x)=1$.

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