# Why is the event $\{X_{(j)} \le x_i\}$ is equivalent to the event $\{Y_i \ge j\}$?

From Statistical Inference by Casella and Berger:

Let $$X_1, \dots X_n$$ be a random sample from a discrete distribution with $$f_X(x_i) = p_i$$, where $$x_1 \lt x_2 \lt \dots$$ are the possible values of $$X$$ in ascending order. Let $$X_{(1)}, \dots, X_{(n)}$$ denote the order statistics from the sample. Define $$Y_i$$ as the number of $$X_j$$ that are less than or equal to $$x_i$$. Let $$P_0 = 0, P_1 = p_1, \dots, P_i = p_1 + p_2 + \dots + p_i$$.

If $$\{X_j \le x_i\}$$ is a "success" and $$\{X_j \gt x_i\}$$ is a "failure", then $$Y_i$$ is binomial with parameters $$(n, P_i)$$.

Then the event $$\{X_{(j)} \le x_i\}$$ is equivalent to the event $$\{Y_i \ge j\}$$

Can someone explain why these two are equivalent?

$$\{X_{(j)} \le x_i\} = \{s \in \text{dom}(X_{(j)}) : X_{(j)}(s) \le x_i\}$$

$$\{Y_i \ge j\} = \{s' \in \text{dom}(Y_i) : Y_i(s') \ge j\}$$

I'm having trouble understanding how these random variable functions show this equivalence.

The following statements are equivalent for every $$\omega\in\Omega$$:

• $$\omega\in\{X_{(j)}\leq x_i\}$$
• $$X_{(j)}(\omega)\leq x_i$$
• $$|\{k\in\{1,\dots,n\}\mid X_k(\omega)\leq x_i\}|\geq j$$
• $$Y_i(\omega)\geq j$$
• $$\omega\in\{Y_i\geq j\}$$

Looking at the first and the last bullet we conclude that: $$\{X_{(j)}\leq x_i\}=\{Y_i\geq j\}$$

If $$X_{(j)}\le x_i$$, then clearly all the other smaller order statistics $$X_{(1)},\ldots, X_{(j-1)}$$ are also less than or equal to $$x_i$$ (since they are less than or equal to $$X_{(j)}$$). So at least $$j$$ of the order statistics are less than or equal to $$x_i$$, or in other words, $$Y_i \ge j$$.

Conversely, if $$Y_i \ge j$$, then by definition at least $$j$$ of the order statistics are less than or equal to $$x_i$$. This implies that $$X_{(j)} \le x_i$$. (Otherwise, $$X_{(j)}$$ is greater than $$x_i$$, and then so are all the greater order statistics, so at most $$j-1$$ order statistics are less than or equal to $$x_i$$, a contradiction.)

• So are the events $\{X_{(j)} \le x_i\}$ and $\{Y_i \ge j\}$ just single point sets? – Oliver G May 30 at 12:49