1. Statistics
Alan D. Smith

2. An Experiment ...

3. An Experiment 100 Pennies Random Toss ...
How many heads do you expect?
How many ways could they land?
What is the “probability” of getting all heads?
What is the “probability” of getting 50 heads & 50 tails?

4. Analysis
Consider 4 pennies
Consider 100 pennies !!!

5. Who Cares about Pennies?
Consider a situation where a manufacturer is concerned about market share. Historically, 10% of the population uses the product. A market survey of 1,000 people reveals that 120 people in the survey use the product. Has market share increased? Should we expand production? How certain are you?

6. Discrete Probability Distributions
Chapter 5

7. WEEK’S GOALS
TO DEFINE THE TERMS PROBABILITY DISTRIBUTION AND RANDOM VARIABLE.
TO DISTINGUISH BETWEEN A DISCRETE AND CONTINUOUS PROBABILITY DISTRIBUTION.
TO CALCULATE THE MEAN, VARIANCE, AND STANDARD DEVIATION OF A DISCRETE PROBABILITY DISTRIBUTION.
TO DESCRIBE THE CHARACTERISTICS OF THE BINOMIAL DISTRIBUTION.
TO INTRODUCE THE POISSON DISTRIBUTION

8. RANDOM VARIABLES
Definition: A random variable is a numerical value determined by the outcome of an experiment. (A quantity resulting from a random experiment that, by chance, can assume different values).
EXAMPLE : Consider a random experiment in which a coin is tossed three times. Let X be the number of heads. Let H represent the outcome of a Head and T the outcome of a Tail.

9. EXAMPLE (Continued) Sample Number of Space Heads = X TTT TTH 1 THT THH
HHT
HHH 1 2 3
Sample
Space
Number of
Heads = X

10. PROBABILITY DISTRIBUTIONS
Definition: A probability distribution is a listing of all the outcomes of an experiment and their associated probabilities.

11. Discrete Distribution

12. CHARACTERISTICS OF A PROBABILITY DISTRIBUTION
The probability of an outcome must always be between 0 and 1.
EXAMPLE: P(0 head) = 0.125, P(1 head) = 0.375, etc. in the coin tossing experiment.
The sum of the probabilities of all mutually exclusive outcomes is always 1.
P(0) + P(1) + P(2) + P(3) = 1

13. DISCRETE RANDOM VARIABLE
Definition: A discrete random variable is a variable that can assume only certain clearly separated values resulting from a count of some item of interest.
EXAMPLE: Let X be the number of heads when a coin is tossed 3 times. Here the values for X are: 0, 1, 2 and 3.
EXAMPLE: Let X be the number of start-up businesses that fail given a starting set of The values for X are: 0, 1, …, 99, 100.

14. CONTINUOUS RANDOM VARIABLE
Definition: A continuous random variable is a variable that can assume one of an infinitely large number of values (within certain limitations).
EXAMPLE: (a) Average GPA for all Students
(b) Income
(c) ROI

15. THE MEAN OF A DISCRETE PROBABILITY DISTRIBUTION
The mean reports the central location of the data.
The mean is the long-run average value of the random variable.
The mean of a probability distribution is also referred to as its expected value, E(X).
The mean is a weighted average.

16. MEAN of a PROBABILITY DISTRIBUTION (continued)
The mean of a discrete probability distribution is computed by the formula:
where m (Greek letter, mu) represents the mean and P(X) is the probability of the various outcomes X.

17. The Variance (Standard Deviation) of a Discrete Probability Distribution
The variance measures the amount of spread or variation of a distribution.
The variance of a discrete distribution is denoted by the Greek letter s2 (sigma squared).
The standard deviation is obtained by taking the square root of s2.

18. The Variance (Standard Deviation)- (continued)
The variance of a discrete probability distribution is computed from the formula:
The standard deviation is computed from s = (s 2)1/2 (sigma).

19. EXAMPLE
U-Rent-It is a car rental agency specializing in renting cars to families who need an additional car for a short period of time. Pat Padgett, President, has studied her records for the last 20 weeks and reports the following number of cars rented per week.

20. EXAMPLE (Continued)
Does the above data qualify as a probability distribution?
Convert the number of cars rented per week to a probability distribution. This is shown on the next slide.

21. EXAMPLE (continued)

22. EXAMPLE (continued) Compute the mean number of cars rented per week.
The mean m = E(X) = S[XP(X)] =
Compute the variance of the number of cars rented per week.
The variance s2 = S[(X - m)2P(X)] =

23. CALCULATIONS FOR m IN TABLE FORM

24. CALCULATIONS FOR s2 in TABLE FORM

25. BINOMIAL PROBABILITY DISTRIBUTION
The binomial distribution has the following characteristics:
An outcome of an experiment is classified into one of two mutually exclusive categories - success or failure.
The data collected are the results of counts.
The probability of a success stays the same for each trial. So does the probability of failure.
The trials are independent, meaning that the outcome of one trial does not affect the outcome of any other trial.

26. BINOMIAL PROBABILITY DISTRIBUTION (continued)
To construct a binomial distribution, we let
n be the number of trials.
r be the number of observed successes.
p be the probability of success on each trial.
q be the probability of failure, found by q =1-p.
4! = 4*3*2*1
0! = 1! = 1

27. EXAMPLE
The Department of Labor Statistics for the state of Kentucky reports that 20% of the workforce in Treble County is unemployed. A sample of 14 workers is obtained from the county. Compute the following probabilities (Binomial):
three are unemployed. (n = 14, p = 0.2).
P(r = 3) = ?

28. P(r = 3) USING THE FORMULA
Given : (r = 3, n = 14, p = 0.2, q = 0.8)
Recall
so,
P(r = 3) =
=BINOMDIST(3, 14, 0.2, FALSE) via EXCEL

29. EXAMPLE (continued) at least one of the sampled workers is unemployed.
P(r ³ 1) =
at most two of the sampled workers are unemployed.
P(r £ 2) =
three or more are unemployed.
P(r ³ 3) =

30. Return to Who Cares about Pennies?
Consider a situation where a manufacturer is concerned about market share. Historically, 10% of the population uses the product. A market survey of 1,000 people reveals that 120 people in the survey use the product. Has market share increased? Should we expand production? How certain are you?

31. BINOMIAL PROBABILITY DISTRIBUTION
To construct a binomial distribution, we let
n be the number of trials. (1000)
r be the number of observed successes. ( )
p be the probability of success on each trial. (0.1)
q be the probability of failure, 1-p. (0.90)
P(120) + P(121) + … + P(1000) = =
1 - P(r<119) = Probability of 120 or more.

32. (Note: Binomial n = 6, p = 0.05) EXAMPLE
A company manufactures ball bearings to be used on mountain bikes. It is known that 5% of the diameters of the bearings will be outside the accepted limit (defective). If 6 bearings are selected at random, what is the probability that: Exactly zero will be defective? Exactly one? Exactly two? Exactly three? Exactly four? Exactly five? Exactly six?
(Note: Binomial n = 6, p = 0.05)

33. EXAMPLE (continued) Note that the binomial conditions are met:
There is a constant probability of success (0.05).
There is a fixed number of trials (6).
The trials are independent. (Why?)
There are only two possible outcomes (a bearing is either defective or nondefective).

34. EXAMPLE (continued) How would you use this table?
Binomial Probability Distribution for n = 6 and p = 0.05
How would you use this table?

35. MEAN & VARIANCE OF THE BINOMIAL DISTRIBUTION
The mean is given by
The variance is given by

36. EXAMPLE
For the previous example regarding defective bearings, recall that:
p = 0.05 and n = 6.
Hence, m = np =
s2 = np(1 - p) =

37. POISSON PROBABILITY DISTRIBUTION
The binomial distribution of probabilities will become more and more skewed to the right as the probability of success become smaller.
The limiting form of the binomial distribution where the probability of success p is very small and n is large is called the Poisson probability distribution.
The Poisson distribution can be described mathematically using the formula:

38. POISSON PROBABILITY DISTRIBUTION
m (mu) is the arithmetic mean number of occurrences (successes) in a particular interval of time.
e is the constant (base of the Naperian logarithmic system)
x number of occurrences (successes)
P(X) is the required probability.
=POISSON(x, m, FALSE) via EXCEL

39. EXAMPLE
The Maumee Urgent Care facility specializes in caring for minor injuries, colds and flu. For the evening hours of PM the mean number of arrivals is 4.0 per hour. Assume the arrivals follow the Poisson distribution. Compute the following probabilities.
What is the probability of exactly 4 arrivals in an hour?
P(4) =

40. EXAMPLE (continued)
What is the probability of less than 4 arrivals in an hour?
What is the probability of at least 4 arrivals in an hour?