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 that takes on numerical or categorical values associated with the random outcome of an experiment, where one (and only one) numerical or categorical 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 take on a countable number of values.
Number of steps to the top of the Eiffel Tower*
A continuous random variable can take on 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.
0 ≤ p(x) ≤ 1 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
0.30 2
0.21 3
0.15 4
0.11 5
0.07 6
0.05 7
0.04 8
0.02 9 10
0.01
Say a random variable x follows this pattern: p(x) = (0.3)(0.7)x-1
for x > 0.
This table gives the probabilities (rounded to two decimals) 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
E(X) is read as
“expected value of X” or “mean of X”
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 five cents 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 per trial: 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 (S) or Tails (F)
P(H) = 0.5; P(T) = =0.5
Result on a flip doesn’t affect the outcomes of other flips
x heads in 3 coin flips
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 light bulbs in a production line are defective.
What is the probability that 6 of the 10 randomly selected bulbs are defective?
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