Jiaping Wang Department of Mathematical Science 02/04/2013, Monday

1. Chapter 4. Discrete Probability DistributionsSection 4.1. Random Variables and Their Probability Distributions Jiaping Wang Department of Mathematical Science 02/04/2013, Monday

2. Outline Random Variables and Probability Functions Distribution Functions Examples

3. Part 1. Random Variables and Probability Functions

4. Random Variable A random variable or stochastic variable is a variable  whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense). As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability. For example, flip a fair coin, denote X=0 meaning tail, X=1 meaning head. So P(X=0) = P(X=1) = ½ and P(X=n)=0 for n≠ 0 or 1 . Then the probability P(X≤2)=P(X=0)+P(X=1)+P(X=2)=1.

5. Definition 4.1 The basic concept of "random variable" in statistics is real-valued. However, one can consider arbitrary types such as boolean values, categorical variables, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, functions, and processes.  Definition 4.1 A random variable is a real-valued function whose domain is a sample space. The random variable can be either continuous or discrete.

6. Definition 4.2 A random variable X is said to be discrete if it can take on only a finite number – or a countably infinite number – of possible values x. The probability function of X, denoted by p(x), assigns probability to each value x of X so that the following conditions hold: P(X=x)=p(x)≥0; ∑ P(X=x) =1, where the sum is over all possible values of x. The probability function is sometimes called the probability mass function of X to denote the idea that a mass of probability is associated with values for discrete points.

7. Cont. It is often convenient to list the probabilities for a discrete random variable in a table. See following example.

8. Example 4.1 A local video store periodically puts its used movie in a bin and offers to sell them to customers at a reduced price. Twelve copies of a popular movie have just been added to the bin, but three of these are defective. A customer randomly selects two of the copies for gifts. Let X be the number of defective movies the customer purchased. Find the probability function of X and graph the function. Denote the D as the defective, ND as the non-defective. There are two step selections.

9. Cont.

10. Part 3. Distribution Function

11. Definition 4.3 The distribution function F(b) for a random variable X is F(b)=P(X ≤ b); If X is discrete, Where p(x) is the probability function. The distribution function is often called the cumulative distribution function (CDF).

12. Cont. P(X≤0)=0.04, P(X<0)=0, P(X ≤1)=P(X=0)+P(X=1)=0.36 but P(X<1)=0.04 P(X ≤2)=P(X=0)+P(X=1)+P(X=2)=1.0 P(X<2)=P(X=0)+P(X=1)=0.36 P(X>2)=1.0 P(X ≤ 1.5)=P(X ≤ 1.9)=P(X ≤ 1)=0.36.

13. Cont. Note from last example, we can find F(x) is a right-continuous function but not left-continuous, that is Any function satisfies the following 4 properties is a distribution function: 1. 2. 3. The distribution function is a non-decreasing function: if a<b, then F(a)≤ F(b). The distribution function can remain constant, but it can’t decrease as we increase from a to b. 4. The distribution function is right-hand continuous:

14. Example 4.2 A large university uses some of the student fees to offer free use of its health center to all students. Let X be the number of times that a randomly selected student visits the center during a semester. Based on historical data, the distribution function of X is given as Graph F. Verify that F is a distribution function. Find the probability function associated with F.

15. Cont. 1. Because F is zero for all values less than zero, so 2. As F is one for all values larger than 3, 3. As x increases, F(x) either remains constant or increases, it means F(x) is nondecreasing. 4. There are three jumping points: 0, 1, 2 and 3, we can show for each point, F is right-hand continuous. For example, when h 0+, F(2+h)0.95=F(2).