Let \(X\) be a real valued random variable. Prove the Chebychev inequality:
\[
\mathbf{P}\{|X| \geq \lambda\} \leq \frac{\mathbf{E}\{|X|^p \}}{\lambda^p }
\]
for all \(1\leq p <\infty\) as well as its exponential form
\[
\mathbf{P}\{|X| \geq \lambda\} \leq \frac{\mathbf{E}\{e^{k|X|} \}}{e^{k\lambda} }
\]
when the integrals on the right are finite. Here \(k\) and \(\lambda\) are positive constants.

Hint: combine the facts that \[\mathbf{P}\{|X|^p \geq \lambda\} = \mathbf{E} \mathbf{1}_{\{|X|^p \geq \lambda\}}\]

and

\[ \mathbf{1}_{\{|X|^p \geq \lambda\}}\lambda^p \leq \mathbf{1}_{\{|X|^p \geq \lambda\}}|X|^p\] and \[1= \mathbf{1}_{\{|X|^p \geq \lambda\} }+ \mathbf{1}_{\{|X|^p <\lambda\}}\,  . \]