Random selection

Introduction

The usual treatments deal with a single random variable or a fixed, finite number of random variables, considered jointly. However, there are many common applicationsin which we select at random a member of a class of random variables and observe its value, or select a random number of random variables and obtain some function of those selected.This is formulated with the aid of a counting or selecting random variable N , which is nonegative, integer valued. It may be independent of the class selected, or may be relatedin some sequential way to members of the class. We consider only the independent case. Many important problems require optional random variables, sometimes called Markov times . These involve more theory than we develop in this treatment.

Some common examples:

1. Total demand of N customers— N independent of the individual demands.
2. Total service time for N units— N independent of the individual service times.
3. Net gain in N plays of a game— N independent of the individual gains.
4. Extreme values of N random variables— N independent of the individual values.
5. Random sample of size N — N is usually determined by propereties of the sample observed.
6. Decide when to play on the basis of past results— N dependent on past.

A useful model—random sums

As a basic model, we consider the sum of a random number of members of an iid class. In order to have a concrete interpretation to help visualize the formal patterns, we think of the demandof a random number of customers. We suppose the number of customers N is independent of the individual demands. We formulate a model to be used for a variety of applications.

• A basic sequence $\{X_n:0\le n\}$ [Demand of n customers]
• An incremental sequence $\{Y_n:0\le n\}$ [Individual demands]
  These are related as follows:
  $$X_n=\sum _{k=0}^n{Y}_k\text{for}n\ge 0\text{and}{X}_n=0\text{for}n<0Y_n=X_n-X_{n-1}\text{for}\text{all}n$$
• A counting random variable N . If $N=n$ then n of the Y _k are added to give the compound demand D (the random sum)
  $$D=\sum _{k=0}^N{Y}_k=\sum _{k=0}^{\infty }{I}_{\{N=k\}}{X}_k=\sum _{k=0}^{\infty }{I}_{\left\{k\right\}}\left(N\right)X_k$$