1. Probability Distributions Finite Random Variables

2. Probability Distributions  Recall a Random Variable was: numerical value associated with the outcome of an experiment that was subject to chance Examples:  The number of heads that appear when flipping three coins  The sum obtained when two fair dice are rolled  Value of an attempted loan workout

3. Finite Random Variables  Types of random variables: Finite Random Variable:  Can only assume a countable number of values (i.e. they can all be listed)  Examples Flipping a coin three times & recording # of heads Rolling a pair of dice & recording the sum on each roll

4. Finite Random Variables  Types of Random Variables (cont): Continuous Random Variable:  Can assume a whole range of values which cannot be listed:  Examples: Arrival times of customers using an ATM at the start of the hour Time between arrivals Length of service at an ATM machine  We’ll first look at finite random variables

5. Finite Random Variables  Ex: Suppose we flip a coin 3 times and record each flip. Let X be a random variable that records the # of heads we get.  What are the possible values of X?

6. Finite Random Variables  X can be 0, 1, 2, or 3  Notice that we can list all possible values of the random variable X  This makes X a finite random variable

7. Finite Random Variables  Recall Told us the probability the random variable X (upper case) assumed a certain value x (lower case)  List all possible values of x (lower case) and their probabilities in table

8. Finite Random Variables  Notice each value of the random variable has one probability associated with it  For a finite random variable we call this the probability mass function  Abbreviated (p.m.f.)

9. Finite Random Variables  Why a function? Each value of the random variable has exactly one probability assigned to it Since this is a function, the following are equivalent for a finite random variable:

10. Finite Random Variables  Like any function, the p.m.f. for a finite random variable has several properties: Domain: discrete numbered values (e.g. {0, 1, 2,3} ) Range: Sum:

11. Finite Random Variables  If we graph a p.m.f., should be a histogram with sum of all heights equal to 1  Each bar height corresponds to P(X=x)

12. Finite Random Variables  Cumulative Distribution Function Abbreviated c.d.f. Determines probability of all events occurring up to and including a specific event

13. Finite Random Variables  Ex: Find from our coin problem. Do the same for and  Sol:

14. Finite Random Variables  Notice: IntervalF X (x) = P(X≤x) (-∞,0)0 [0,1)0.125 [1,2)0.125 + 0.375=0.500 [2,3)0.125+0.375+0.375=0.875 [3,∞)0.125+0.375+0.375+0.125=1

15. Finite Random Variables  The last table is describing a piece-wise function:

16. Finite Random Variables  The graph of the c.d.f.

17. Finite Random Variables  Notice for the c.d.f. of a finite random variable Graph is a step-wise function Domain is all real #s Graph never decreases Approaches 1 as x gets larger

18. Finite Random Variables  Ex. Use the sample space for rolling two dice to graph.

19. Finite Random Variables  Soln.

20. Finite Random Variables  Ex. Find in the previous example. Find in the previous example. Find in the previous example.  Soln.

21. Finite Random Variables  Remember: Cumulative distribution function adds probabilities up to and including a certain value  Ex. Find in the previous example. Find in the previous example. Find in the previous example.

22. Finite Random Variables  Soln.

23. Finite Random Variables Sample c.d.f.

24. Finite Random Variables  Notice the graph starts at height 0  Notice the graph ends at height 1  The graph “steps” to next height  The size of each “step” corresponds to the value of the p.m.f. at that x-value

25. Finite Random Variables  The bullet on the last slide gives us the ability to: p.m.f. c.d.f

26. Finite Random Variables  Binomial Random Variable Special type of finite r.v. Collection of Bernoulli Trials Bernoulli Trial is an experiment with only 2 possible outcomes Each trial is independent

27. Finite Random Variables  Excel has a built in Binomial R.V. function  BINOMDIST

28. Finite Random Variables  Ex: Suppose we flip a biased coin whose probability of landing heads is 0.7. Let’s say we do this 3 times and record each flip. Let X be a random variable that records the # of heads we get.  This is an example of a binomial random variable

29. Finite Random Variables  In Excel:  This is the p.m.f Value of Random Variable How many times you perform the experiment Probability of success on each trial FALSE (p.m.f) ; TRUE (c.d.f)

30. Finite Random Variables  In Excel:  This is the c.d.f

31. Finite Random Variables  Ex. Historically, 83% of all students pass a particular class. If there are 34 students in the class, what is the probability that exactly 28 will pass? What is the probability that at least 28 students pass? What is the probability that at most 28 students pass?

32. Finite Random Variables  What is the probability that exactly 28 students pass? Soln: 0.1760

33. Finite Random Variables  What is the probability that at least 28 students pass? Soln: 1 - 0.3545 0.6455

34. Finite Random Variables  What is the probability that at most 28 students pass? Soln: 0.5305

35. Finite Random Variables  Sample p.m.f.

36. Finite Random Variables  Mean of a finite random variable same as expected value in project 1 add product of each value and it’s respective probability

37. Finite Random Variables  Ex. Historically, 83% of all students pass a particular class. If there are 34 students in the class, find the mean number of students that will pass. Soln. Approximately 28.22 students will pass (Two ways this can be found)

38. Finite Random Variables  Method 1:  Use BINOMDIST to make p.m.f.  Then use:

39. Finite Random Variables  Method 2:  There is a special formula that ONLY works for random variables that are binomially distributed: