Lecture 1: Binomial & Poisson Distribution

In today's lecture, you have to take note of the following points:

1. What is a random variable? What is probability distribution?

2. Under what condition will the random variable follow Binomial distribution?

3. What is the probability distribution function for Binomial distribution?

What is a random variable? What is probability distribution?

Two important characteristics of a random variables are reflected in its name. " Variable" tells us that it is a value that can change. "Random" tells us that the change of the value is random, i.e. there is no way to predict the value.

There are two types of random variables: discrete and continuous. If you can count the possible values taken by the random variable, it is discrete. Otherwise, it is continuous.

A random variable can take a certain set of possible values. For each of these value, there is a certain probability associated. The way that the probability is distributed among these possible values is called the probability distribution. It can be summarized in terms of a formula or in a table form, which is called the probability distribution function of the random variable.

Under what condition will the random variable follow Binomial distribution?

• There are n independent trials. (Independence means that the outcome of any trials will not affect the outcome of rest of the trials. )
• For each trial, there are two possible outcomes, one success and one failure.
• The probability of success is constant in each trial.

Remark: It is very important to know the condition for each type of distribution. It helps you to determine the right distribution to use when given a problem.

3. What is the probability distribution function for Binomial distribution?

There are two important values to take note of for each binomial distribution: the no. of trials, n, and the probability of success, p. If X~B(n, p)

\[P(X=r)=^{n}\textrm{C}_{r}p^{r}q^{n-r}\]

After this lecture, you can attempt Tutorial 14A Q1