Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 16 Random Variables

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Expected Value: Center A random variable assumes a value based on the ___________ of a random event. We use a capital letter, like X, to denote a __________ variable. A ______________ value of a random variable will be denoted with a lower case letter, in this case x.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Expected Value: Center (cont.) There are two types of random variables: ____________ random variables can take one of a finite number of distinct outcomes. Example: _____________ random variables can take any numeric value within a range of values. Example:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Expected Value: Center (cont.) A _______________________ for a random variable consists of: The collection of all ____________ values of a random variable, and the ______________ that the values occur. Of particular interest is the value we expect a random variable to take on, notated μ (for population mean) or E(X) for _______________.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Expected Value: Center (cont.) The expected value of a (discrete) random variable can be found by summing the products of each _______________ by the ____________ that it occurs: Note: Be sure that ________ possible outcome is included in the sum and verify that you have a ________ probability model to start with.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide First Center, Now Spread… For data, we calculated the standard deviation by first computing the deviation from the mean and squaring it. We do that with random variables as well. The variance for a random variable is: The standard deviation for a random variable is:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To calculate the mean and stadard deviation using the TI Enter random variable values in ___ and enter the associated probabilities in ___. Calculate _________________. Note that Slide 16- 7

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Just Checking For a particular mechanical problem, 75% of the time the problem can be fixed by a simple $60 repair job. However, the rest of the time the complex job will cost an additional $140. A)Define the random variable and construct the probability model. B)What is the expected value of the cost of this repair? C)What does that mean in this context? D)What is the standard deviation of cost? Slide 16- 8

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Assignment P. 381 #1-8, Slide 16- 9

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 16 Random Variables (2)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide More About Means and Variances Adding or subtracting a constant from data shifts the _______ but doesn’t change the __________ or _____________________: E(X ± c) = ________ Var(X ± c) = _______ Example: Consider everyone in a company receiving a $5000 increase in salary.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide More About Means and Variances (cont.) In general, multiplying each value of a random variable by a constant multiplies the _______ by that constant and the _________ by the _______ of the constant: E(aX) = ______Var(aX) = _________ Example: Consider everyone in a company receiving a 10% increase in salary.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide More About Means and Variances (cont.) In general, The mean of the sum of two random variables is the ___________________. The mean of the difference of two random variables is the _______________________. E(X ± Y) = _____________ If the random variables are independent, the variance of their sum or difference is always the ____________________. Var(X ± Y) = ___________________

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Continuous Random Variables Random variables that can take on any value in a range of values are called ___________ random variables. Continuous random variables have means (__________values) and variances. We won’t worry about how to calculate these means and variances in this course, but we can still work with models for continuous random variables when we’re given the _____________.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Continuous Random Variables (cont.) Good news: nearly everything we’ve said about how discrete random variables behave is true of continuous random variables, as well. When two independent continuous random variables have ________________, so does their sum or difference. This fact will let us apply our knowledge of Normal probabilities to questions about the _____ or __________ of independent random variables.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What Can Go Wrong? Probability models are still just models. Models can be _________, but they are not _________. Question probabilities as you would data, and think about the ________________ behind your models. If the ________ is _______, so is everything else.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What Can Go Wrong? (cont.) Don’t assume everything’s _________. Watch out for variables that aren’t ___________: You can add expected values for _____ two random variables, but you can only add variances of _____________ random variables.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What Can Go Wrong? (cont.) Don’t forget: Variances of independent random variables ____. Standard deviations _______. Don’t forget: Variances of independent random variables ____, even when you’re looking at the _____________ between them. Don’t forget: Don’t write independent instances of a random variable with notation that looks like they are the ______ variables.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What have we learned? We know how to work with random variables. We can use a probability model for a _______ random variable to find its ________________ and ______________________. The mean of the sum or difference of two random variables, discrete or continuous, is just the ____ _________________ of their means. And, for independent random variables, the variance of their sum or difference is always the _____ of their variances.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What have we learned? (cont.) Normal models are once again special. Sums or differences of Normally distributed random variables also follow _____________.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Practice The American Veterinary Association claims that the annual cost of medical care for dogs averages $100, with a standard deviation of $30, and for cats averages $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) If the difference in costs can be described by a Normal model, what’s the probability that medical expenses are higher for someone’s dog than for her cat? d) If a person has two dogs and a cat, what’s the probability that their total annual expenses will be greater than $400? Slide

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Assignment P. 382 #21-23, 25, 28, 32, 33, 35, 38, 39 Slide