Chernoff and Markov tail bounds

Let \(\xi\) be a continuous random variable.

1. Prove that if \(\mathbf { E } e ^ { \lambda \xi }\) is finite then

\[ P ( \xi \geqslant x ) \leqslant e ^ { - \lambda x } \mathbf { E } e ^ { \lambda \xi } , \quad \lambda > 0 .\tag{45} \]

2. Prove that if \(\mathbf { E } | \boldsymbol { \xi } | ^ { m }\) is finite then

\[ P ( | \xi | \geqslant x ) \leqslant x ^ { - m } \mathbf { E } | \xi | ^ { m } , x > 0 , m > 0 .\tag{46} \]