From the statistical inference of Casella and Berger:

Leave $ X_1, points X_n $ Be a random sample of a discrete distribution.

with $ f_X (x_i) = p_i $, where $ x_1 lt x_2 lt dots $ they are possible

values of $ X $ in ascending order. Leave $ X _ {(1)}, dots, X _ {(n)} $

denotes the order statistics of the sample. Define $ Y_i $ as the number of $ X_j $ that are less than or equal to

$ x_i $. Leave $ P_0 = 0, P_1 = p_1, dots, P_i = p_1 + p_2 + dots + p_i $.

Yes $ {X_j le x_i } $ it's a "success" and $ {X_j gt x_i } $ it is a "failure", then $ Y_i $ it 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 anyone explain why these two are equivalent?

$ {X_ {(j)} le x_i } = {s in text {sun} (X _ {(j)}): X _ {(j)} (s) le x_i } $

$ {Y_i ge j } = {s & # 39; in text {sun} (Y_i): Y_i (s & # 39;) ge j } $

I have trouble understanding how these functions of random variables show this equivalence.