Understanding the Expected Value of a Binomial Distribution

The expected value, often denoted as E(X), of a binomial distribution is a fundamental concept in probability theory and statistics. It represents the average outcome if we were to repeat an experiment many times under identical conditions. This article will explore the derivation and significance of this expected value, providing a clear explanation of the underlying mathematical principles.

Derivation of the Expected Value of a Binomial Distribution

Let's begin by evaluating the sum:

$$S sum_{i0}^{n} i binom{n}{i} p^i (1-p)^{n-i}$$

This expression represents the expected value of a binomial distribution. To understand this, let's break it down step by step.

Step 1: Utilizing a Combinatorial Identity

First, we rewrite the term i binom{n}{i} using the identity:

$$i binom{n}{i} n binom{n-1}{i-1}$$

Substituting this into the original sum, we get:

$$S sum_{i0}^{n} n binom{n-1}{i-1} p^i (1-p)^{n-i}$$

Step 2: Adjusting the Summation Index

Notice that the sum starts from i 0, but since i 0 contributes nothing, we start the summation from i 1:

$$S n sum_{i1}^{n} binom{n-1}{i-1} p^i (1-p)^{n-i}$$

Next, we change the index of summation by letting j i - 1. This shifts the index from i to j. When i 1, j 0; when i n, j n - 1. The sum becomes:

$$S n sum_{j0}^{n-1} binom{n-1}{j} p^{j 1} (1-p)^{n-1-j}$$

Step 3: Simplifying the Summation

The sum (sum_{j0}^{n-1} binom{n-1}{j} p^{j 1} (1-p)^{n-1-j}) can be rewritten as:

$$S n p sum_{j0}^{n-1} binom{n-1}{j} p^j (1-p)^{n-1-j}$$

Notice that the sum inside is the binomial expansion of ((p (1-p))^{n-1} 1^{n-1} 1). Therefore, we have:

$$S n p cdot 1 n p$$

Significance of the Expected Value

The expected value (np) of a binomial distribution is the average number of successes in (n) independent trials, where each trial has a success probability (p). This formula

is the expression for the expected value mean of an experiment repeated (n) independent times. The experiment has two possible outcomes, following a binomial distribution, where the probability of an event of interest occurring is (p), and the probability of its complement event is (1-p).

In other words, given the probability (p) of a binomial event occurring and (n) independent repeats of the experiment, the formula will give you the likely average number of times that the event with the probability (p) of occurring each time would occur, given (n) repeats of the experiment.

What is ( binom{n}{i} )?

The binomial coefficient (binom{n}{i}) can be viewed as the number of distinct ways to choose (i) items out of (n) items, where the ordering is not important. It is defined as:

( binom{n}{i} frac{n!}{i!(n-i)!} )

where (n! n(n-1)(n-2)cdots 3 cdot 2 cdot 1).

Thus, the value of the sum is given by:

( boxed{np} )