The equalities should be Jensen inequality, Markov Inequality, Tchebychev Inequality.

Be detail!

1. Applying Inequalities, $2 p$ Let $X$ be a r.v. and $? > 0$ Using one the inequalities considered in the last lecture prove that $P [X ? ?] ? \frac{E [ e ^{tX} ]}{e ^{? t}}, ? t > 0$

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The equalities should be Jensen inequality, Markov Inequality, Tchebychev Inequality.

Be detail!

1. Applying Inequalities,

2 p

Let

X

be a r.v. and

? > 0

Using one the inequalities considered in the last lecture prove that

P [X ? ?] ? \frac{E [ e ^{tX} ]}{e ^{? t}}, ? t > 0

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