1. Discrete Distributions
Chapter 7
Discrete Distributions

2. Random Variable -
A numerical variable whose value depends on the outcome of a chance experiment

3. Two types: Discrete – count of some random variable
Continuous – measure of some random variable

4. Discrete and Continuous Random Variables
A random variable is discrete if its set of possible values is a collection of isolated points on the number line.
Possible values of a discrete random variable
A random variable is continuous if its set of possible values includes an entire interval on the number line.
Possible values of a continuous random variable
We will use lowercase letters, such as x and y, to represent random variables.

5. Examples 1. Experiment: A fair die is rolled
Random Variable: The number on the up face
Type: Discrete
2. Experiment: A coin is tossed until the 1st head turns up
Random Variable: The number of the toss that the 1st head turns up

6. 3. Experiment: Choose and inspect a number of parts
Random Variable: The number of defective parts
Type: Discrete
4. Experiment: Measure the voltage in a outlet in your room
Random Variable: The voltage
Type: Continuous
5. Experiment: Observe the amount of time it takes a bank teller to serve a customer
Random Variable: The time

7. Discrete Probability Distribution
Gives the probabilities associated with each possible x value
Usually displayed in a table, but can be displayed with a histogram or formula

8. Discrete probability distributions
3)For every possible x value,
0 < P(x) < 1.
4) For all values of x,
S P(x) = 1.

9. Suppose you toss 3 coins & record the number of heads
Suppose you toss 3 coins & record the number of heads. What is the sample space?

10. The Random Variable X is defined as …
The number of heads tossed
# of heads
X=0 TTT
X=1 HTT,THT,TTH
X=2 HHT,HTH,THH
X=3 HHH

11. Create a probability distribution.
Create a probability histogram. X
P(X)

12. Create a probability distribution.X
P(X)
Now we can use the probability distribution table to answer questions about the variable X.
What is the probability of getting exactly 2 heads?
P(X=2) =.375
What is the probability of getting at least 2 heads?
P(X>2) = P(X=2) + P(X=3) = .5
What is the probability of getting at least one head?
P(X>1) = P(X=1) + P(X=2) + P(X=3) = .875

13. Why does this not start at zero?
Let x be the number of courses for which a randomly selected student at a certain university is registered.
X P(X) ?
P(x = 4) =
P(x < 4) =
What is the probability that the student is registered for at least five courses?
Why does this not start at zero?
.25
.14
P(x > 5) = .61
.39

14. Mean and Variance of Discrete Random Variables

15. Probability Distributions are also described by measures of central tendency and variability.
The MEAN of a discrete random variable X is the average of the possible outcomes of X WITH the weights (probabilities). Other names for the MEAN are the WEIGHTED AVERAGE or the EXPECTED VALUE.

16. Probability Distributions are also described by measures of central tendency and variability.
The VARIANCE is an average of the squared deviation of the values of the variable X from its mean. The STANDARD DEVIATION is the square root of the variance.
𝒙 𝒊 − 𝝁 𝒙 𝟐 𝒑 𝒊

17. Formulas for mean & variance
Found on formula card!

18. What is the mean and standard deviations of this distribution?
Let x be the number of courses for which a randomly selected student at a certain university is registered. X
P(X)
What is the mean and standard deviations of this distribution?
m = & s =

19. Find the mean and standard deviation for the number of heads out of 3 tosses.X
P(X)
m = & s = .866

20. A fair game is one where the cost to play EQUALS the expected value!
Here’s a game:
If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If he gets a 7, he wins $5. The cost to roll the dice one time is $3. Is this game fair?
A fair game is one where the cost to play EQUALS the expected value!

21. NO, since m = $1.944 which is less than it cost to play ($3).
If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If he gets a 7, he wins $5. The cost to roll the dice one time is $3. Is this game fair?
What is the random variable X?
Make a probability distribution table. X
P(X) 7/9 1/6 1/18
NO, since m = $1.944 which is less than it cost to play ($3).

22. What is the random variable X? _____________________
Let’s play a game. A player pays $5 and draws a card from a deck. If he draws the ace of hearts, he is paid $100. For any other ace, he is paid $10, and for any other heart, he is paid $5. If he draws anything else, he gets nothing. Would you be willing to play?
What is the random variable X? _____________________
What are the values for the random variable? ____________
Outcome
Payout (X)
Probability
E(X)
Deviation

23. An insurance company offers a “death and disability” policy that pays $10,000 when you die or $5000 if you are permanently disabled. It charges a premium of on $50 a year for this benefit. Is the company likely to make a profit selling such a plan?

24. Suppose that the death rate in any year is 1 out of every 1000 people, and that another 2 out of 1000 suffer some kind of disability.
Policyholder
Outcome
Payout X
Probability
P(x)
Death
$10,000
Disability
$5000
Neither $0

25. E (X) = Expected value = μ = = $10,000( ) + $5000( ) + $0( )
Policyholder
Outcome
Payout X
Probability
P(x)
Death
$10,000
Disability
$5000
Neither $0
E (X) = Expected value = μ =
= $10,000( ) + $5000( ) + $0( )
= $10 + $10 + $0
= $20

26. Standard deviation (X) = √149,600 = $386.78
Policyholder
Outcome
Payout X
Probability
P(x)
Death
$10,000
Disability
$5000
Neither $0
Deviation
(x – μ)
(10,000 – 20) = 9980
(5000 – 20) = 4980
(0 – 20) = - 20
Var(x) = 99802( ) ( ) + (-20)2 ( ) = 149,600
Standard deviation (X) = √149,600 = $386.78

27. Linear function of a random variable
The mean is changed by addition & multiplication!
If x is a random variable and a and b are numerical constants, then the random variable y is defined by
and
The standard deviation is ONLY changed by multiplication!

28. Let x be the number of gallons required to fill a propane tank
Let x be the number of gallons required to fill a propane tank. Suppose that the mean and standard deviation is 318 gal. and 42 gal., respectively. The company is considering the pricing model of a service charge of $50 plus $1.80 per gallon. Let y be the random variable of the amount billed. What is the mean and standard deviation for the amount billed?
m = $ & s = $75.60

29. Linear combinations Just add or subtract the means!
If independent, always add the variances!

30. The mean of the sum of two random variables is the sum of the means.
E(X + Y) = E(X) + E(Y)
The mean of the difference of two random variables is the difference of the means.
E(X - Y) = E(X) - E(Y)
If the random variable are independent, the variance of their sums OR difference is always the sum of the variances.
Var(X + Y) = Var(X) + Var(Y)

31. What is the mean and standard deviation of:SD X 10 2 Y 20 5
What is the mean and standard deviation of:
a) 3X
Mean = 30; standard deviation = 6
b) Y + 6
Mean = 26; standard deviation = 5
c) X + Y
Mean = 30; standard deviation = = 5.39
d) X - Y
Mean = -10; standard deviation = = 5.39
e) X1 + X1
Mean = 20; standard deviation = = 2.83

32. A nationwide standardized exam consists of a multiple choice section and a free response section. For each section, the mean and standard deviation are reported to be
mean SD MC FR
If the test score is computed by adding the multiple choice and free response, then what is the mean and standard deviation of the test?
m = & s =

33. Example
Suppose x is the number of sales staff needed on a given day. If the cost of doing business on a day involves fixed costs of $255 and the cost per sales person per day is $110, find the mean cost (the mean of x or mx) of doing business on a given day where the distribution of x is given below.

34. We need to find the mean of y = 255 + 110x
Example continued
We need to find the mean of y = x

35. We need to find the variance and standard deviation of y = 255 + 110x
Example continued
We need to find the variance and standard deviation of y = x