Raising the bar

Chapter 5. Discrete Random Variables and Probability Distributions

Unit 5.13. Distribution of the binomial variable

Case $latex n = 3$. When you need to recall what a binomial variable is, you start with "it is a sum" and then disclose the rest. Let us derive the distribution of $latex {S_3}$. The idea is to list all possible combinations of values of $latex {X_1},\,{X_2},\,{X_3}$, find the corresponding values of $latex {S_3}$ and then calculate their probabilities. This is done in two steps. Remember that the objective is to get the "values plus probabilities" table.

In mathematical exposition ideas sometimes are not stated. The presumption is that you get the idea after reading the proof. In your mind, ideas should always go first.

Step 1. The first step is realized in Table 5.8.

Table 5.8. Possible outcomes and their probabilities.

Here we used the multiplication rule. For example, the probability $latex p{q^2}$ in the second row is obtained like this: by independence

$latex P({X_1} = 1,\,{X_2} = 0,\,{X_3} = 0) = P({X_1} = 1)P({X_2} = 0)P({X_3} = 0) = p{q^2}.$

Step 2. The next step is to collect equal values of $latex {S_3}$. This is done using the additivity rule. For example, the events (1,0,0), (0,1,0) and (0,0,1) in rows 2--4 are mutually exclusive and therefore

$latex P({S_3} = 1) = P((1,0,0) \cup (0,1,0) \cup (0,0,1))$

$latex = P(1,0,0) + P(0,1,0) + P(0,0,1) = 3p{q^2}.$

Thus the result is Table 5.9.

Table 5.9. Distribution of $latex {S_3}$.

Exercise 5.8. Using Table 5.9 find $latex E{S_3},\,\,V({S_3})$.

Case of general $latex n$. The first thing to understand is that $latex {S_n}$ takes all integer values $latex x$ from 0 to $latex n$, inclusive, and for each of these values we need to figure out the probability $latex P({S_n} = x)$ (that's what Table 5.9 is about in case $latex n = 3$).

Step 1. The equation $latex {S_n} = x$ tells us that in a sequence $latex ({X_1},...,{X_n})$ of zeros and unities there are $latex x$ unities and $latex n - x$ zeros. The probability of ONE such sequence is $latex {p^x}{q^{n - x}}$ by the multiplication rule.

Step 2. We have to find out how many such sequences exist for the given $latex x$. This question can be rephrased as follows: in how many different ways $latex x$ places (where the unities will be put) can be chosen out of $latex n$ places? Since the order doesn't matter, the answer is: in $latex C_x^n$ ways. Because each of these ways has the same probability $latex {p^x}{q^{n - x}}$ and they are mutually exclusive, by the additivity rule

$latex P({S_n} = x) = C_x^n{p^x}{q^{n - x}},\,\,x = 0,...,n. \qquad (\textrm{Eq. 5.15})$

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