5.1 notes introduction to random variables and probability distributions
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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