Ordered statistic CDF written as a multinomial distribution

Question

so I'm reading Casella Berger and am stuck on a problem regarding ordered statistics. the problem 5.26 part a) states ``Let U be a random variable that counts the number $X_{1},...,X_{n}$ less than or equal to u, and V be a r.v. counting the number $X_{1},...,X_{n}$ greater than or equal to u, and less than or equal to v. show that $(U,V,n-U-V) \sim \text{Multinomial}(F_{X}(u),F_{X}(v)-F_{X}(u),1-F_{X}(v))$

my work: P($U\leq u \leq V \leq v)$ is what we wish to find. and this is over 3 intervals (0,u), (u,v), (n,v-u). but i'm having a bit of trouble starting this problem.

any hint to deepen my understanding is very appreciated.

Answer 1

Recall the meaning of a multinomial distribution. For the sake of simplicity, suppose that there is a multinomial trial in which each trial has three mutually exclusive possible outcomes, which occur with probabilities $p_1, p_2, p_3$, respectively, and $p_1 + p_2 + p_3 = 1$. (Thus far we are simply looking at a generalization of Bernoulli/binomial trials to the case where there are three instead of two outcomes.) Then there is a vector-valued random variable $\boldsymbol Y = (Y_1, Y_2, Y_3)$ that describes the number of outcomes observed of each type, and we have $$\Pr[\boldsymbol Y = (y_1, y_2, y_3)] = \frac{n!}{y_1! y_2! y_3!} p_1^{y_1} p_2^{y_2} p_3^{y_3}, \quad y_1 + y_2 + y_3 = n.$$ Now, for the set of iid random variables $\boldsymbol X = (X_1, X_2, \ldots, X_n)$, we can regard each one as an independent "Tri-noulli" trial (actually this is a categorical random variable with three categories), for which $Y_i$ counts the number of $X_j$s that satisfy the $i^{\rm th}$ criterion, because the criteria are mutually exclusive; either:

1. $X_j \le u$ which implies $p_1 = \Pr[X \le u] = F_X(u)$;
2. $u < X_j \le v$ which implies $p_2 = \Pr[u < X \le v] = F_X(v) - F_X(u)$;
3. $v < X_j$ which implies $p_3 = \Pr[X > v] = 1 - F_X(v)$.

More formally, $$U = Y_1 = \sum_{j=1}^n \mathbb 1 (X_i \le u), \\ V = Y_2 = \sum_{j=1}^n \mathbb 1(u < X_i \le v), \\ n - U - V = Y_3 = \sum_{j=1}^n \mathbb 1(X_i > v).$$ Then the resulting distribution is clearly multinomial with the described parameters.