Sections 6.1 & 6.2 Completed Notes

1

Chapter 6 - Random VariablesSections 6.1 & 6.2

Random variables -

· Take on values based on the outcome of a random event

· Can be discrete or continuous (discrete we can list all the outcomes, continuous - think about the Normal model)

· Probability models list all possible values and probabilities that they occur - keep in mind this looks different for discrete and continuous variables

EXAMPLE:Suppose I have decided to set up a Vegas night to raise money for a school club. I have set up a game where you roll two dice and the sum of the two dice determines your prize. Suppose if you roll snake eyes (double ones) or boxcars (double sixes) you win $20. However, if you roll a sum of 4, 6, 8, or 10 you win only $3 and a sum of 3, 5, 7, 9, or 11 you win $2. How much can I expect to pay out?

Random variable =

Values of the random variable =

Probabilities of those values =

What should we charge so that we make a profit in the long run?

Sections 6.1 & 6.2 Completed Notes

2

Expected Value of Discrete Random Variables

Just take each value of the random variable, multiply by the probability of that value, and add them up!

A man buys a racehorse for $20,000 and enters it in two races. He plans to sell the horse afterward, hoping its value will jump to $100,000. If it wins one of the races, it will be worth $50,000. If it loses both races, it will be worth only $10,000. The man believes there's a 20% chance that the horse will win the first race and a 30% chance it will win the second one. Assuming that the two races are independent events, find the man's expected profit.

Sections 6.1 & 6.2 Completed Notes

3

Standard Deviation and Variance of Random VariablesFrom our previous example:Suppose I have decided to set up a Vegas night to raise money for a school club. I have set up a game where you roll two dice and the sum of the two dice determines your prize. Suppose if you roll snake eyes (double ones) or boxcars (double sixes) you win $20. However, if you roll a sum of 4, 6, 8, or 10 you win only $3 and a sum of 3, 5, 7, 9, or 11 you win $2. How much can I expect to pay out?

Let's calculate the standard deviation of this random variable...

Standard Deviation and Variance of Random Variables(for Discrete Variables)

Sections 6.1 & 6.2 Completed Notes

4

An insurance policy costs $100 and will pay policyholders $10,000 if they suffer a major injury (resulting in hospitalization) or $3,000 if they suffer a minor injury (resulting in lost time from work). The company estimates that each year 1 in every 2000 policyholders may have a major injury, and 1 in 500 a minor injury.

a) Create a probability model for the company's profit on policy.

b) What's the company's expected profit on this policy?

c) What's the standard deviation?

Continuous Random VariablesContinuous Random Variables can take on infinite numbers of values (sometimes all real numbers!) so finding the probability that a random variable takes on a particular value is impossible.

Instead we find the probability that a random variable falls within a given interval (i.e. less than some value, greater than some value, or between two values)

Does this sound familiar?

Sections 6.1 & 6.2 Completed Notes

5

Normal ProbabilitiesThe Normal distribution is the most common distribution of a continuous random variable.

In the first unit we learned about Normal curves and used these to describe the distribution of some random variables.

We can also use the Normal model as a probability model as well.

The probability that a Normal random variable falls within a particular interval is the SAME as the area under the corresponding Normal curve for that same interval - i.e. it's exactly what we've done under the disguise "Probability"

FOR EXAMPLE:The scores for SAT II English Exams in a given year were Normally distributed with a mean of 450 and a standard deviation of 120. What is the probability of scoring higher than a 600?

Sections 6.1 & 6.2 Completed Notes

6

The scores on SAT tests in a particular year were Normally distributed with a mean of 1026 and a standard deviation of 209. What is the probability that a random student who took the SAT test that year scored between an 800 and 1200?

Properties of Means and Variances of Random Variables(for Discrete AND Continuous Random Variables)

If X and Y are independent:

For two random variables, X and Y:

Sections 6.1 & 6.2 Completed Notes

7

Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of each of these variables:

a) X - 20

b) 0.5Y

c) X + Y

d) X - Y

e) 2Y1 + Y2

Mean SDX 80 12

Y 12 3

If two independent continuous random variables have Normal models, so does their sum or difference - this is very useful when we are doing calculations of probabilities for Normal models that involve more than one independent Normal random variables

Normal breeds Normal...

Sections 6.1 & 6.2 Completed Notes

8

The American Veterinary Association claims that the annual cost of medical care for dogs and cats follows Normal models. For dogs the average cost is $100 with a standard deviation of $30 and for cats the average $120 with a standard deviation of $35.

a) What's the expected difference in the cost of medical care for dogs and cats?

b) What's the standard deviation of that difference?

c) What's the probability that medical expenses are higher for someone's dog than for her cat?

Atalocalhighschool,theheightofmalestudentsfollowaNormaldistributionwithmeanof68inchesandastandarddeviationof3.4in.TheheightsoffemalestudentsalsofollowaNormalmodelwithmean63inchesandstandarddeviationof2.8in.What'stheprobabilitythatthehomecomingking(amalestudent)ismorethan2inchestallerthanthehomecomingqueen(afemalestudent)?

Sections 6.1 & 6.2 Completed Notes

9

p.353#s1-9odd,13,14,18,19,23,25,27-30all

p.378#s37,39-41all,43,45,49,51,57-59all,

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