Chapter 6 Summary

A random variable. In Chapter 5 we began to look at random variables. A random variable takes on numerical values that result from outcomes of an experiment.

Discrete Versus Continuous. In Chapter 6, we learned that a random variable may be either discrete or continuous. A discrete random variable is one that takes on a fixed number of possible values whereas a continuous random variable can take on any value within an interval (and thus it has infinitely many values). Another way to differentiate between the two is to realize that a discrete random variable is something that we count whereas a continuous random variable is something that we measure. Our goal in this chapter was to begin to understand discrete probability models and to study one type of discrete probability model, the binomial probability model. A discussion for probability models for continuous random variables will begin in Chapter 7.

Modeling a Probability Distribution. A discrete probability model results when a random variable has only a fixed number of potential outcomes. The discrete probability model includes both the possible outcomes that a random phenomenon can take on as well as each outcome’s corresponding probability. In every discrete probability model, each probability must be a number between 0 and 1, inclusive, and the probabilities must sum to one.

Determining probabilities for a discrete random variable can often be simplified with the use of a tree diagram. The discrete probability model is commonly summarized using a table.

The Meaning of Expected Value. Often, we are interested in measuring the expected value of a random variable. The expected value, μ_x, is the average value that the random variable takes on, in the long run If the random variable, X, is discrete, then the expected value is found by multiplying each value of the random variable by its probability and then summing these products. This is expressed mathematically as follows:

\( \mu_x = \sum x \cdot P(x) \).

The standard deviation, σ_x, of a discrete random variable, X, can be found using the formula

\( \sigma_x = \sqrt{(x - \mu_x)^2P(x)} \).

The Binomial Setting. A common (and important) type of a discrete probability model is the binomial probability model. A binomial experiment is one that involves the repetition of an experiment (called a trial) and one that must satisfy the following four conditions:

1. each trial has only two outcomes, success and failure;
2. the probability of success, p, remains the same from trial to trial;
3. the trials are independent; and
4. the experiment is performed a fixed (n) number of times.

When a binomial experiment is performed, the number X of successes is typically recorded. The probability of having k successes in n trials can be computed using the formula,

\( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \).

If X is a binomial random variable, then the expected value of X, \( \mu_x = n \times p \) is np and the standard deviation is \( \sigma_x = \sqrt{n \times p \times (1-p)} \).