1. Discrete Random Variables and Probability Distributions3
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3.1
Random Variables
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3. Z = the length of the fish in inches
Random Variables
Informally: A rule for assigning a real number to each outcome of a random experiment
A fish is randomly selected from a lake. Let the random variable
Z = the length of the fish in inches
A voter is randomly selected and asked if he or she supports a candidate running for office. Let the random variable

4. Random Variables
Such a rule of association is called a random variable—a variable because different numerical values are possible and random because the observed value depends on which of the possible experimental outcomes results (Figure 3.1).
A random variable
Figure 3.1

5. Random Variables
Definition

6. Random Variables
Random variables are customarily denoted by uppercase letters, such as X and Y, near the end of our alphabet.
In contrast to our previous use of a lowercase letter, such as x, to denote a variable, we will now use lowercase letters to represent some particular value of the corresponding random variable.
The notation X (s) = x means that x is the value associated with the outcome s by the rv X.

7. Example 3.1
When a student calls a university help desk for technical support, he/she will either immediately be able to speak to someone (S, for success) or will be placed on hold (F, for failure). With = {S, F}, define an rv X by X (S) = X (F) = 0 The rv X indicates whether (1) or not (0) the student can immediately speak to someone.

8. Random Variables - observations
A random variable is a function.
A random variable is not a probability.
Random variables can be defined in practically any way. Their values do not have to be positive or between 0 and 1 as with probabilities.
Random variables are typically named using capital letters such as X, Y , or Z. Values of random variables are denoted with their respective lower-case letters. Thus, the expression
X = x
means that the random variable X has the value x.

9. Discrete Random Variables
Definition A random variable is discrete if its range is either finite or countable.
Chapter 4 – Continuous random variables

10. Probability Mass Functions
Definition Let X be a discrete random variable and let R be the range of X. The probability mass function (abbreviated p.m.f.) of X is a function f : R → ℝ that satisfies the following three properties:

11. Distribution
Definition The distribution of a random variable is a description of the probabilities of the values of variable.

12. Example
A bag contains one red cube and one blue cube. Consider the random experiment of selecting two cubes with replacement. The two cubes we select are called the sample. The same space is S = {RR, RB, BR, BB}, and we assume that each outcome is equally likely. Define the random variable X = the number of red cubes in the sample

13. Example X(RR) = 2, X(RB) = 1, X(BR) = 1, X(BB) = 0 Values of X:
Values of the p.m.f.:

14. Example
The Distribution Table Probability Histogram Formula

15. Uniform Distribution
Definition Let X be a discrete random variable with k elements in its range R. X has a uniform distribution (or be uniformally distributed) if its p.m.f. is

16. Example
Consider the random experiment of rolling a fair six-sided die. The sample space is S ={1, 2, 3, 4, 5, 6} Let the random variable X be the number of dots on the side that lands up. Then (i.e. X has a uniform distribution)

17. Cumulative Distribution Function
Definition The cumulative distribution function (abbreviated c.d.f.) of a discrete random variable X is

18. Example
Consider the random experiment of rolling two fair six-sided dice and calculating the sum. Let the random variable X be this sum.

19. Example
Consider the random experiment of rolling two fair six-sided dice and calculating the sum. Let the random variable X be this sum.

20. Mode and Median
Definition The mode of a discrete random variable X is a value of X at which the p.m.f. f is maximized. The median of X is the smallest number m such that

21. Example
Mode = 1
Median = 1 since

22. Random Variables
Definition

23. Example 3.3
Example 2.3 described an experiment in which the number of pumps in use at each of two six-pump gas stations was determined. Define rv’s X, Y, and U by X = the total number of pumps in use at the two stations Y = the difference between the number of pumps in use at station 1 and the number in use at station 2 U = the maximum of the numbers of pumps in use at the two stations

24. Example 3.3
cont’d
If this experiment is performed and s = (2, 3) results, then X((2, 3)) = = 5, so we say that the observed value of X was x = 5.
Similarly, the observed value of Y would be y = 2 – 3 = –1, and the observed value of U would be u = max(2, 3) = 3.