This problem is asking to calculate the Probability Mass Function (PMF) for the order statistics of a set of $m$ independent and identically distributed random variables $X_1, X_2, \ldots, X_m$, which are drawn from a binomial distribution with parameters $n$ and $p$.

Explanation of the Problem:

• Binomial Distribution: The random variables $X_i$ follow $\text{Bin}(n, p)$, meaning: $P(X_i = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \ldots, n.$

• Order Statistics: For a sequence of random variables $X_1, X_2, \ldots, X_m$, the $i$-th order statistic $X_{(i)}$ represents the $i$-th smallest value in the sorted list. This requires calculating the PMF of $X_{(i)}$.

Steps to Solve:

1. Joint Distribution: Consider the joint distribution of $X_1, X_2, \ldots, X_m$.
2. Permutation and Sorting: Compute the PMF of the $i$-th order statistic $X_{(i)}$, taking into account how many of the $X_j$'s are less than or equal to a given value.

Let me calculate the detailed solution now.To solve this, we need to derive the PMF of the order statistic $X_{(i)}$ for the given sequence of independent binomial random variables $X_1, X_2, \ldots, X_m$.

Step-by-Step Solution:

1. Probability Distribution of $X_i$

Each $X_i$ is binomially distributed: $P(X_i = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \ldots, n.$

2. Defining Order Statistics

Let $X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(m)}$ be the ordered values. The $i$-th order statistic $X_{(i)}$ is the $i$-th smallest value among $X_1, X_2, \ldots, X_m$.

To find the PMF of $X_{(i)}$, say $P(X_{(i)} = k)$, we consider:

• $k$ of the $X_j$'s are $\leq x$,
• $(m-k)$ of the $X_j$'s are $> x$,
• The $i$-th position corresponds to the cutoff value.

3. General Formula for Order Statistics PMF

The PMF of $X_{(i)}$ is: $P(X_{(i)} = k) = \binom{m}{i} P(X \leq k)^{i} P(X > k)^{m-i}.$ Here:

• $P(X \leq k)$: Cumulative probability of the Binomial up to $k$,
• $P(X > k) = 1 - P(X \leq k)$.

4. Cumulative Probability for Binomial

The cumulative probability $P(X \leq k)$ for $X \sim \text{Bin}(n, p)$ is: $P(X \leq k) = \sum_{j=0}^k \binom{n}{j} p^j (1-p)^{n-j}.$

Similarly: $P(X > k) = 1 - \sum_{j=0}^k \binom{n}{j} p^j (1-p)^{n-j}.$

5. Final Expression for $P(X_{(i)} = k)$

Substitute $P(X \leq k)$ and $P(X > k)$ into the formula: $P(X_{(i)} = k) = \binom{m}{i} \left( \sum_{j=0}^k \binom{n}{j} p^j (1-p)^{n-j} \right)^i \left( 1 - \sum_{j=0}^k \binom{n}{j} p^j (1-p)^{n-j} \right)^{m-i}.$

This gives the PMF for the $i$-th order statistic of the binomial variables.

Would you like a numerical example or additional clarification on any of these steps? Let me know!

Follow-Up Questions:

1. What are the properties of order statistics in general distributions?
2. How does the formula change for continuous distributions?
3. Can you compute $P(X_{(i)} \leq x)$ instead of the PMF?
4. How would the result change if the distribution were uniform instead of binomial?
5. How is the concept of order statistics applied in real-world problems?

Tip:

Always check if the random variables are identically distributed before directly applying formulas for order statistics. For non-identical distributions, adjustments are required.