Discrete Populations (Probability Models)

As discussed above discrete random variables have a finite (or listable) range. What are their probability models? It's easy. In fact, you knew this before taking this course. Right! The probability model of a discrete random variable  is its range and associated probabilities.

For example, in the first roll in the game of craps (roll a pair of dice) let X = the sum of the upfaces. Then the range of X is 2, 3, 4, ... ,12. Now ASSUME that the dice are fair. Upon recalling the picture of the sample space, it is easy to determine the probability model of X. For example, the probability that X = 3 means the probability that a (1,2) or a (2,1) comes up which is (1/36) + (1/36) = 2/36. Using the same reasoning for the other range items, we obtain the probability model for X:

Range            2    3    4    5    6    7    8    9   10   11   12
Probabilities  1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

Instead of a table, how about a picture of the probability model? Just plot the probabilities (vertical) versus the range (horizontal). Here is a crude plot:

                                  .    :    .
                        .    :    :    :    :    :    .
              .    :    :    :    :    :    :    :    :    :    .
           ---+---------+---------+---------+---------+---------+---
            2.0       4.0       6.0       8.0      10.0      12.0

Alas, two more items. A little notation here is useful. For a discrete random variable X, let p(x) denote the probability that X assumes the value x. Often p(x) is called the probability mass function.  For example, in the dice problem above. We will denote the probability that X is 7 by p(7); hence, p(7) = 6/36.

Note that, in general for any discrete random variable, p(x) is a fraction and the sum of all the p(x) (over the range of X) is 1.

Although the term probability model makes sense here, it is often not used in practice. Usually we call the probability model of X, the distribution  of X. It is confusing since we used the term distribution with sampling distributions of Chapter 1. We will sort this out later.

Exercise 5.2.1  

1.

Let X denote the number spun on a fair spinner with the numbers 1, 2, and 3 on it. Determine the probability model of X.

2.

In the last problem, suppose we spin the the spinner twice. Let S be the sum of the numbers spun. The range of S is 2, 3, 4, 5, 6. Use a tree diagram to determine the probability model of S.

3.

Repeat the last problem if the spinner is spun 3 times.

4.

Let X denote the number of aces in a 2 card hand drawn without replacement from a well shuffled standard deck 0f 52 cards. Then the range of X is {0, 1, 2}. Use a tree diagram to determine the probability model of X.

5.

Repeat the last problem under sampling with replacement.

6.

Let X denote the number of hearts in a 2 card hand drawn without replacement from a well shuffled standard deck 0f 52 cards. Then the range of X is {0, 1, 2}. Use a tree diagram to determine the probability model of X.

7.

In the urn problem (with the balls well mixed ) discussed above, let Z denote the number of red balls in the sample (without replacement) of size 3. Use a tree diagram to determine the probability model of Z. What is the range of Z? Is it discrete or continuous?