Random variables inequality

Question

Let $X_1, X_2, ... , X_n$ be non-negative random variables on a probability space $(\Omega , P)$, each distributed as a random variable X, and let $S_n = \sum_{j=1}^{n} X_j$.

By considering appropriate events, deduce that

$$P (X_j \leq \frac{\lambda}{n}\;\text{for all j} ) \leq P (S_n \leq \lambda ) \leq P (X_j \leq \lambda \; \text{for all j} )$$

Hence show that if the $X_j$ are mutually independent, then

$$P^n (X \leq \frac{\lambda}{n}) \leq P (S_n \leq \lambda ) \leq P^n (X_j \leq \lambda )$$

By taking complements, show that

$$P^n (X \leq \frac{\lambda}{n}) \geq 1 - nP (X > \frac{\lambda}{n})$$

In a previous part of the question, I verified that $max\; X_j \leq S_n \leq n \;max X_j $ , however I could not see how to bring that forward to deduce the second pair of inequalities. Any help would be much appreciated.

Answer 1

The second pair of inequalities follows immediately from the first by the fact that $X_1,...,X_n$ are iid. You have, for example,

$$P(X_j \leq \lambda/n \text{ for all $j$}) = P(\cap_j\{X_j \leq \lambda/n \}) = \prod_j P(X_j \leq \lambda/n) = P^n(X \leq \lambda/n).$$

To get the first pair of inequalities, simply note that the first event in question implies the second, and the second event implies the third (by the non-negativity of $X_j$). Then use the fact that if $A$ implies $B$, then $P(A) \leq P(B)$.