1. Probability Distribution for Discrete Random Variables

2. Introduction
Probabilities assigned to various outcomes in S in turn determine probabilities associated with the values of any particular random variable X.
The probability distribution of X gives how the total probability of 1 is distributed among the various X values.

3. Probability mass function
To give the probability distribution or probability mass function (pmf) of a discrete rv X, we give the values that X can be, and the probability of taking on those values.

4. Example 1
Consider rolling a pair of die. Then S contains the set of 36 ordered pairs {(1,1), (1,2), …, (6,6)}
Let X=sum of values on the die, e.g. X(4,3)=7.
What is the probability distribution of X?

5. Example 2
The Cal Poly Department of Statistics has a lab with six computers reserved for statistics majors. Let X denote the number of these computers that are in use at a particular time of day.
The table on the next slide gives the probability of each value.

6. Example 2 (continued)
We can use elementary properties of probability to calculate other probabilities of interest. As examples, x 1 2 3 4 5 6
p(x)
.05
.10
.15
.25
.20

7. Example 3
Six lots of components are ready to be shipped by a certain supplier. The number of defective components in each lot is as follows:
One of these lots will be picked at random to be shipped.
Then p(0)=3/6=1/2; p(1)=1/6; p(2)=2/6=1/3
Lot 1 2 3 4 5 6
# defective

8. A parameter of a probability distribution
The pmf of a Bernoulli rv X is of the form:
Each value of yields a different pmf is called a parameter of the distribution.

9. Definition of a parameter
Suppose p(x) depends on a quantity that can be assigned any one of several possible values, with each value giving a different probability distribution. The quantity is called a parameter of the distribution.
The collection of probability distributions for different values of the parameter is called a family of probability distributions.

10. Example 4
We observe the gender of each newborn child at a hospital until a boy is born. Let p=P(B), and assume that successive births are independent. Then
may be appropriate for this situation, but
may be more appropriate if we are looking for the first child with RH-positive blood. This is the family of geometric distributions.

11. Definition of the cumulative distribution function
The cumulative distribution function (cdf) F(x) of a discrete rv with pmf p(x) is defined for every number x by

12. Example 5
For x 1 2 3 4 5 6
p(x)
.05
.10
.15
.25
.20

13. Going from cdf to pmf
The probability mass function is given by the size of the jumps of the cumulative distribution function.

14. Using the cdf For any two numbers a and b with ,
where represents the largest possible X value that is strictly less than If only integer values are possible for a and b, then