5.1 Notes

Introduction to Random Variables and Probability Distributions

Discrete Random Variable –

i.e. #

Continuous Random Variable –

i.e.

Ex. 1 Which of the following random variables are discrete and which are continuous?

a) Time it takes a student to register for classes

b) The number of “bad checks” drawn on a checking account

c) The amount of gasoline needed to drive your car 200 miles.

d) The amount of voters in the last local election.

Probability Distribution – Same as a relative frequency distribution

1.

2.

Boredom ToleranceTest Scores

Score # of subjects Probability

0 1400

1 2600

2 3600

3 6000

4 4400

5 1600

6 400

a) Find the probability of receiving each score on the boredom test.b) Make a histogram of the results from part a)

c) If Top Notch Clothing Company needs to hire someone with a score of 5 or 6 to operate a fabric press machine, what is the probability that a person chosen at random will score 5 or 6 on the test?

Ex. 2 Dr. Fidget developed a test to measure boredom tolerance. He administered it to a group of 20,000 adults between the ages of 25 and 35. The possible scores were 0, 1, 2, 3,, 4, 5, 6, with 6 indicating the highest tolerance for boredom. The test results for this group are shown in the table.

Mean of a probability distribution:

Standard Deviation of a probability distribution:

Both of the previous values are found more easily by putting x-values in L1 and putting the corresponding probability in L2 and then computing 1 VarStat, L1, L2

Ex. 3 Are we influenced to buy a product because we saw an ad on TV? National Infomercial Marketing Association determined the number of times buyers of a product watched a TV infomercial before purchasing the product. The results are as follows:

*This category was 5 or more, but will be treated as 5 in this example.

a) Is the previous a probability distribution? Justify.

b) Find the mean and standard deviation of the distribution.

For Continuous Data, use midpoint of the range for then x-values.

# of Times BuyersSaw Infomercial

1 2 3 4 5*

% of Buyers 27% 31% 18% 9% 15%

Assignmentp. 178 #2, 3, 6, 8, 11, 12, 13

Linear Combinations of Random Variables

Let x1 and x2 be independent random variables with respective means μ1 and μ2, and variances For the linear combination W = ax1 + bx2, the mean, variance, and standard deviation are as follows:

μW =

22

21 and

2

W

W

Ex. 3 Let x1 and x2 be independent random variables with respective means μ1 = 75 and μ2 = 50, and standard deviations σ1 = 16 and σ2 = 9.

a) Let L = 3 + 2x1. Compute the mean, variance, and standard deviation of L.

b) Let W = x1 + x2. Find the mean, variance, and standard deviation of W.

c) Let W = x1 – x2. Find the mean, variance, and standard deviation of W.

d) Let W = 3x1 – 2x2. Find the mean, variance, and standard deviation of W.

μL =

2 L L

μW =

2 W W

μW =

2 W W

μW =

2 W W

Assignmentp.181 #14-16