1. Discrete Distributions
Chapter 7
Discrete Distributions

2. Random Variable Numerical variable whose value depends on chance
Capital letters – X or Y

3. Two types: Discrete: X = a count of some random variable
Continuous: X = a measurement of some random variable

4. Discrete Probability Distribution
Gives probabilities for each possible X value
Displayed in a table, histogram, or formula

5. Discrete Probability Distributions
For every possible x value,
0 < P(x) < 1
For all values of x,
Σ P(x) = 1

6. Suppose you toss 3 coins & record the number of heads.
The random variable X would be…
Create a probability distribution.
Draw the histogram for this distribution.
The number of heads tossed x
P(x)

7. 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
.39
P(X > 5) = .61

8. Formulas for Mean & Variance
Found on green sheet!

9. Suppose you toss 3 coins & record the number of heads.
Find the mean and standard deviation for the number of heads out of 3 tosses. x
P(x)
μ = 1.5 & σ = .866

10. What is the mean and standard deviation 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 deviation of this distribution?
μ = & σ =

11. 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 profit! x
P(x) 7/9 1/6 1/18
No – μ = $1.944, which is less than the cost of playing.

12. Linear Transformation of a Random Variable
The mean is changed by addition & multiplication
The standard deviation is ONLY changed by multiplication
If X and Y are random variables, and Y = a + bX, then:

13. 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 are 318 gal and 42 gal, respectively.
The company is considering 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?
μ = $ & σ = $75.60

14. Linear Combinations of Random Variables
Just add or subtract the means
If:
Then:
If the variables are independent, always add the variances

15. A nationwide standardized test 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 total test score?
μ = & σ =