Proving that E[L] ≤ E[L|L>n], where where the inequality is strict if P(L≤ n)> 0

Question

How do I show that E[L] ≤ E[L|L>n] for any discrete random variable L, where the inequality is strict if P(L≤ n)> 0. E[X] represents the expected value of the random variable X.

Answer 1

Use the facts that $ELI_{L>n} \geq nP\{L>n\} \, \,$ (1) and

$ELI_{L\leq n}\leq n P\{L\leq n\} \,\, $ (2).

We get $ELI_{L\leq n} P\{X>n\}\leq n P\{X>n\}P\{L\leq n\} \leq ELI_{L>n} P\{L\leq n\}$.

This gives $ELI_{L\leq n} P\{X>n\} \leq ELI_{L>n} (1-P\{L> n\})$. Simple algebraic manipulation now gives the desired inequality. If $P\{L\leq n\} >0$ then the inequality in (1) is strict and you get strict inequality.