In this lesson, and some of the lessons that follow in this section, we'll be looking at specially named discrete probability mass functions, such as the geometric distribution, the hypergeometric distribution, and the poisson distribution. As you can probably gather by the name of this lesson, we'll be exploring the well-known binomial distribution in this lesson.
The basic idea behind this lesson, and the ones that follow, is that when certain conditions are met, we can derive a general formula for the probability mass function of a discrete random variable \(X\). We can then use that formula to calculate probabilities concerning \(X\) rather than resorting to first principles. Sometimes the probability calculations can be tedious. In those cases, we might want to take advantage of cumulative probability tables that others have created. We'll do exactly that for the binomial distribution. We'll also derive formulas for the mean, variance, and standard deviation of a binomial random variable.
- To understand the derivation of the formula for the binomial probability mass function.
- To verify that the binomial p.m.f. is a valid p.m.f.
- To learn the necessary conditions for which a discrete random variable \(X\) is a binomial random variable.
- To learn the definition of a cumulative probability distribution.
- To understand how cumulative probability tables can simplify binomial probability calculations.
- To learn how to read a standard cumulative binomial probability table.
- To learn how to determine binomial probabilities using a standard cumulative binomial probability table when \(p\) is greater than 0.5.
- To understand the effect on the parameters \(n\) and \(p\) on the shape of a binomial distribution.
- To derive formulas for the mean and variance of a binomial random variable.
- To understand the steps involved in each of the proofs in the lesson.
- To be able to apply the methods learned in the lesson to new problems.