Exercise C.3.7

Let SS be a sample space, and let XX and X′X' be random variables such that X(s)≥X′(s)X(s) \ge X'(s) for all s∈Ss \in S. Prove that for any real constant tt,

Pr⁡{X≥t}≥Pr⁡{X′≥t} \Pr\{X \ge t\} \ge \Pr\{X' \ge t\}

Both of them, expanded, are:

Pr⁡{X≥t}=∑s∈S:X(s)≥tPr⁡{s}Pr⁡{X′≥t}=∑s∈S:X′(s)≥tPr⁡{s} \Pr\{X \ge t\} = \sum_{s \in S:X(s) \ge t}\Pr\{s\} \\ \Pr\{X' \ge t\} = \sum_{s \in S:X'(s) \ge t}\Pr\{s\}

Each term of the second sum is present in the first sum, because X(s)≥X′(s)X(s) \ge X'(s). This makes them at least equal. There might be additional terms in the first sum (when X(s)≥t>X′(s)X(s) \ge t > X'(s). Thus it can also be greater.