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I am confused about the following problem in my probability textbook (solution provided below it). We're given that the random variable $X$ is less than the random variable $Y$ at the given value. But I do not understand how they can say that $\{y\leq w\} \subset \{x \leq w\}$.

Where does that fact come from?

Show that if $X(\zeta) \leq Y(\zeta)$ for every $\zeta \in \mathcal{I}$, then $F_X(w) \geq F_Y(w)$ for every $w$.

The proof is given as:

If $Y(\zeta_i) \leq w$, then $X(\zeta_i) \leq w$ because $X(\zeta_i) \leq Y(\zeta_i)$.

Hence, $$\{Y \leq w\} \subset \{X \leq w\} \qquad \qquad \mathbb{P}(Y \leq w) \leq \mathbb{P}(X \leq w).$$ Therefore, $$F_Y(w) \leq F_X(w).$$

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    If $y(\zeta)\leqslant w$ and $x(\zeta)\leqslant y(\zeta)$ then $x(\zeta)\leqslant$ $____$. – 2017-02-19

1 Answers 1

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We are told that for every outcome, $\xi$, in the sample space, then $x(\xi)

That means for any such outcome where $y(\xi)\leq w$, then it is also true that $x(\xi)\leq w$.   (However, the converse is not necessarily so.)

Therefore every outcome in the event $\{\xi:y(\xi)\leq w\}$ is an outcome in the event $\{\xi:x(\xi)\leq w\}$.

Thus by definition of subset, $\{\xi:y(\xi)\leq w\}\subseteq\{\xi:x(\xi)\leq w\}$ .

We quite usually abbreviate this as $\{y\leq w\}\subseteq\{x\leq w\}$.

Then the properties required of a probability measure mean that: $\mathsf P(y\leq w)\leq \mathsf P(x\leq w)$