Download A random variable is a numerical value associated with each

A random variable is a numerical value associated with each outcome.
The random variable is usually represented by x.
A discrete variable has a finite number of values.
This is covered here in chapter 6.
Examples
• The number of heads on a certain number of flips
• The number of 1’s rolled on a certain number of dice.
• The number of questions correct on a test.
A continuous variable has an infinite number of values.
This is covered in chapters 7 and 8.
Examples
• Speed
• Height
• Weight
A discrete probability distribution lists all possible outcomes and their associated probabilities.
Same as a relative frequency distribution.
• Probabilities must be between 0 and 1.
• Sum of the probabilities equals 1.
Since it has all outcomes, it is a population.
Examples
1. The number of heads on five flips of a coin.
2. The sum of the top sides when you roll two dice.
3. The number of correct guesses on a ten question four choice multiple choice test.
4. A public speaker gave the following number of speeches per week.
Number of speeches per week
0
1
2
3
4
5
Number of weeks
18
126
66
36
45
9
Create a probability distribution.
Find the probability of giving 2 speeches in a week.
Find the probability of giving 3 or 4 speeches in a week.
Find the probability of giving at least 1 speech in a week.
Find the probability of giving less than 4 speeches in a week.
5. The number of commercial shown in a half hour children’s show has the following
probability distribution.
Number of commercials
5
6
7
8
9
Probability
0.20
0.25
0.38
0.10
0.07
Find the probability of 5 or 6 commercials.
Find the probability of at most 8 commercials.
Find the probability of more than 6 commercials.
Descriptive Statistics
Graph
Use the histogram with the outcomes and probability.
Center
Use the mean, which in probability is also called the expected value.
•
x P( x )
Spread
Use the standard deviation and variance.
•
(x
•
x 2 P( x)
•
) 2 P( x)
2
Unusual values are defined as before.
Examples
1. Find the mean and standard deviation for the example 1 above. What do they mean?
2. Find the mean and standard deviation for the example 2 above. What do they mean?
3. Find the mean and standard deviation for the example 3 above. What do they mean?
4. Find the mean and standard deviation for the example 4 above. What do they mean?
5. Find the mean and standard deviation for the example 5 above. What do they mean?
6. A raffle offers a $1000 prize, a $500 prize, and five $100 prizes. A thousand tickets are sold
at $3 apiece. What is the expectation? What does it mean? Is the raffle fair? Why or why not?
7. A 60 year old buys a $1000 life insurance policy for $60. The probability of a 60 year old to
live to 61 is 0.972. Find the expectation of the policy. What does it mean?