1. Chapter 4: Discrete Random Variables
Statistics
Chapter 4: Discrete Random Variables

2. McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
Where We’ve Been
Using probability to make inferences about populations
Measuring the reliability of the inferences
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

3. McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
Where We’re Going
Develop the notion of a random variable
Numerical data and discrete random variables
Discrete random variables and their probabilities
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

4. 4.1: Two Types of Random Variables
A random variable is a variable hat assumes numerical values associated with the random outcome of an experiment, where one (and only one) numerical value is assigned to each sample point.
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

5. 4.1: Two Types of Random Variables
A discrete random variable can assume a countable number of values.
Number of steps to the top of the Eiffel Tower*
A continuous random variable can assume any value along a given interval of a number line.
The time a tourist stays at the top
once s/he gets there
*Believe it or not, the answer ranges from 1,652 to 1,789. See Great Buildings
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

6. 4.1: Two Types of Random Variables
Discrete random variables
Number of sales
Number of calls
Shares of stock
People in line
Mistakes per page
Continuous random variables
Length
Depth
Volume
Time
Weight
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

7. 4.2: Probability Distributions for Discrete Random Variables
The probability distribution of a discrete random variable is a graph, table or formula that specifies the probability associated with each possible outcome the random variable can assume.
p(x) ≥ 0 for all values of x
p(x) = 1
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

8. 4.2: Probability Distributions for Discrete Random Variablesx
P(x) 1
.30 2
.21 3
.15 4
.11 5
.07 6
.05 7
.04 8
.02 9 10
.01
Say a random variable x follows this pattern: p(x) = (.3)(.7)x-1
for x > 0.
This table gives the probabilities (rounded to two digits) for x between 1 and 10.
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

9. 4.3: Expected Values of Discrete Random Variables
The mean, or expected value, of a discrete random variable is
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

10. 4.3: Expected Values of Discrete Random Variables
The variance of a discrete random variable x is
The standard deviation of a discrete random variable x is
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

11. 4.3: Expected Values of Discrete Random Variables
Chebyshev’s Rule
Empirical Rule
≥ 0
 .68
≥ .75
 .95
≥ .89
 1.00
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

12. 4.3: Expected Values of Discrete Random Variables
In a roulette wheel in a U.S. casino, a $1 bet on “even” wins $1 if the ball falls on an even number (same for “odd,” or “red,” or “black”).
The odds of winning this bet are 47.37%
On average, bettors lose about a nickel for each dollar they put down on a bet like this.
(These are the best bets for patrons.)
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

13. 4.4: The Binomial Distribution
A Binomial Random Variable
n identical trials
Two outcomes: Success or Failure
P(S) = p; P(F) = q = 1 – p
Trials are independent
x is the number of Successes in n trials
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

14. 4.4: The Binomial Distribution
Flip a coin 3 times
Outcomes are Heads or Tails
P(H) = .5; P(F) = 1-.5 = .5
A head on flip i doesn’t change P(H) of flip i + 1
A Binomial Random Variable
n identical trials
Two outcomes: Success or Failure
P(S) = p; P(F) = q = 1 – p
Trials are independent
x is the number of S’s in n trials
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

15. 4.4: The Binomial Distribution
Results of 3 flips
Probability
Combined
Summary
HHH
(p)(p)(p) p3
(1)p3q0
HHT
(p)(p)(q)
p2q
HTH
(p)(q)(p)
(3)p2q1
THH
(q)(p)(p)
HTT
(p)(q)(q)
pq2
THT
(q)(p)(q)
(3)p1q2
TTH
(q)(q)(p)
TTT
(q)(q)(q) q3
(1)p0q3
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

16. 4.4: The Binomial Distribution
The Binomial Probability Distribution
p = P(S) on a single trial
q = 1 – p
n = number of trials
x = number of successes
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

17. 4.4: The Binomial Distribution
The Binomial Probability Distribution
The probability of getting the required number of successes
The probability of getting the required number of failures
The number of ways of getting the desired results
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

18. 4.4: The Binomial Distribution
Say 40% of the class is female.
What is the probability that 6 of the first 10 students walking in will be female?
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

19. 4.4: The Binomial Distribution
A Binomial Random Variable has
Mean Variance Standard Deviation
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

20. 4.4: The Binomial Distribution
For 1,000 coin flips,
The actual probability of getting exactly 500 heads out of 1000 flips is
just over 2.5%, but the probability of getting between 484 and 516 heads
(that is, within one standard deviation of the mean) is about 68%.
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

21. 4.5: The Poisson Distribution
Evaluates the probability of a (usually small) number of occurrences out of many opportunities in a …
Period of time
Area
Volume
Weight
Distance
Other units of measurement
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

22. 4.5: The Poisson Distribution
 = mean number of occurrences in the given unit of time, area, volume, etc.
e = ….
µ = 
2 = 
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

23. 4.5: The Poisson Distribution
Say in a given stream there are an average of 3 striped trout per 100 yards. What is the probability of seeing 5 striped trout in the next 100 yards, assuming a Poisson distribution?
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

24. 4.5: The Poisson Distribution
How about in the next 50 yards, assuming a Poisson distribution?
Since the distance is only half as long,  is only half as large.
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

25. 4.6: The Hypergeometric Distribution
In the binomial situation, each trial was independent.
Drawing cards from a deck and replacing the drawn card each time
If the card is not replaced, each trial depends on the previous trial(s).
The hypergeometric distribution can be used in this case.
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

26. 4.6: The Hypergeometric Distribution
Randomly draw n elements from a set of N elements, without replacement. Assume there are r successes and N-r failures in the N elements.
The hypergeometric random variable is the number of successes, x, drawn from the r available in the n selections.
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

27. 4.6: The Hypergeometric Distribution
where
N = the total number of elements
r = number of successes in the N elements
n = number of elements drawn
X = the number of successes in the n elements
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

28. 4.6: The Hypergeometric Distribution
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables

29. 4.6: The Hypergeometric Distribution
Suppose a customer at a pet store wants to buy two hamsters for his daughter, but he wants two males or two females (i.e., he wants only two hamsters in a few months)
If there are ten hamsters, five male and five female, what is the probability of drawing two of the same sex? (With hamsters, it’s virtually a random selection.)
McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables