1. Discrete Probability Distributions Lecture 1 Sections: 5.1 – 5.2
Chapter 5
Discrete Probability Distributions
Lecture 1
Sections: 5.1 – 5.2

2. Random Variables
Random Variable: A variable (say x) that assumes a single numerical value, determined by chance, for each outcome of a procedure.
Discrete Random Variable: A random variable that assumes values that can be counted.
Continuous Random Variable: A random variable that assumes values that cannot be counted. It has infinitely many values, such that there is no gap between any 2 numbers.

3. Example (Just as before):
1. Number of accidents on the 405 by UCLA.
2. Number of Girls in a family of 3 kids.
3. Amount of Diet Coke consumed in a year .
4. Daily temperature at ELAC.

4. Probability Distribution
Probability Distribution: A graph, table or chart that illustrates the probability for each value of the random variable.
Recall a couple that wants to have 3 children:
BBB
BBG
BGB
BGG
GBB
GBG
GGB
GGG
Probability of Girls
x(Girls)
P(x)

5. What do you notice in this Probability Distribution? x(Girls) P(x)
Probability of Girls
x(Girls)
P(x)
0 ≤ P(x) ≤ 1 for each x.
2. Σ P(x) = 1 for all possible values of x.
Requirements for a Probability Distribution.
1. 0 ≤ P(x) ≤ 1 for each x.
2. Σ P(x) = 1 for all possible values of x.

6. The following are not Probability Distributions. Why? 1. x P(x) 0.1 1
0.1 1
0.2 2
0.3 3
0.4 4
0.5 2. x
P(x)
0.25 1 2
-0.25 3
0.5
These two examples do not fulfill the requirements for a Probability Distribution.
0 ≤ P(x) ≤ 1 for each x.
2. ∑P(x) = 1 for all possible values of x.

7. Example:
Based on past results of games of the NBA Finals, the probability that the finals will last four games is 0.042, five games is 0.246, six games is 0.429, and seven games is Construct a table and determine if the given information is a Probability Distribution. If so, determine the mean, variance and standard deviation.
Let x = Games to Conclude the NBA Finals x
P(x) 4
0.042 5
0.246 6
0.429 7
0.283
Yes, the information on the NBA Finals is a Probability Distribution.
1. 0 ≤ P(x) ≤ 1 for each x.
2. ∑P(x) = 1 for all possible values of x.

8. Mean: μ = ∑[ x ∙ P(x) ]
Variance: σ 2 = ∑[ x2 ∙ P(x) ]–μ2
Standard Deviation: x
P(x) 4
0.042 5
0.246 6
0.429 7
0.283
x P(x) x2
x2 P(x)
0.168 16
0.672
1.23 25
6.15
2.574 36
15.444
1.981 49
13.867
∑[ x ∙ P(x) ] = 5.95, ∑[ x2 ∙ P(x) ] =
μ=5.95, σ2= – (5.95)2= , σ =

9. Maximum Usual Value: μ + 2σ Minimum Usual Value:μ – 2σ
We have found that:
μ=5.95, σ2=0.7305, σ =
Just as before,
Maximum Usual Value: μ + 2σ
Minimum Usual Value:μ – 2σ
μ + 2σ =
μ – 2σ =
What does this tell us about the NBA Finals?
It is unusual for the NBA Finals to last only four games.
We could have just analyzed the probability of the finals ending in four games. P(4 games) = ≤ Thus it is unusual for the NBA Finals to last only four games.

10. 3. Nick’s Barber Shop has 5 chairs for waiting customers
3. Nick’s Barber Shop has 5 chairs for waiting customers. Nick conducted a study and he noticed if the chairs were full, no one would stand and wait. He also indicated that that the probability distribution for the number of waiting customers is as follows: x
P(x)
0.424 1
0.161 2
0.134 3
0.111 4 5
0.077
Complete the table. Also, find the mean and standard deviation.
What can we conclude?

11. Expected Value
When we speak of the mean of a Discrete Random Variable, we want to think of it as the “Expected Value”. What we “Expect” the outcome to be. This is crucial when making decisions.
Therefore, the mean of a Discrete Random Variable is also the Expected Value. E= μ = ∑[ x ∙ P(x) ]
Example: You play the Daily 3,(California Lottery) and you bet $1 on the number What is the expected value of this game?

12. Event x P(x) x∙P(x) Win $499 0.001 0.499 Lose –$1 0.999 –0.999
Example: For this event, there are two simple outcomes. You either win or you lose. Since you want to win with the number 121, there is a probability of winning and a probability of losing. You first have to find the winning payoff. In this case, the winning payoff is $499 to $1. This means that if you win, you win $499 and if you lose, you lose $1. This tells you that you will let x = payoff of the game.
Event x
P(x)
x∙P(x)
Win
$499
0.001
0.499
Lose
–$1
0.999
–0.999
E= –$0.50
This says that in the long run, for each bet of $1, you can expect to lose an average of 50¢. Thus the California Lottery’s “take” is 50%. If E = 0, then the game would be a Fair Game.

13. What does this say about the Insurance Policy?
4. A 30 year old woman wants to purchase Life Insurance. She will pay $200 for a one-year policy that will have coverage of $100,000. The probability of her living through the year is What is her expected value?
Let x = payoff of insurance policy
Event x
P(x)
x∙P(x)
Die
Live
What does this say about the Insurance Policy?

14. 5. In New Jersey’s pick 4 lottery game, you pay 50¢ to select a sequence of four digits, such as If you win by selecting the same sequence of four digits that are drawn, you collect $ What is the expected value?