copyright © 2010 pearson education, inc. chapter 16 random variables

Copyright © 2010 Pearson Education, Inc.

Chapter 16Random Variables

Copyright © 2010 Pearson Education, Inc. Slide 16 - 3

Expected Value: Center

A random variable assumes a value based on the outcome of a random event. We use a capital letter, like X, to denote a

random variable. A particular value of a random variable will be

denoted with the corresponding lower case letter, in this case x.


Expected Value: Center (cont.)

There are two types of random variables: Discrete random variables can take one of a

countable number of distinct outcomes. Example: Number of credit hours

Continuous random variables can take any numeric value within a range of values.

Example: Cost of books this term



A probability model for a random variable consists of: The collection of all possible values of a

random variable, and the probabilities 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 expected value.



The expected value of a (discrete) random variable can be found by summing the products of each possible value by the probability that it occurs:

Note: Be sure that every possible outcome is included in the sum and verify that you have a valid probability model to start with.

E X x P x


Example

On Valentine’s Day, a restaurant offers a special that could save couples money on their romantic dinners. When the waiter brings the check, he’ll also bring four aces from a deck of cards. He’ll shuffle them and lay them out face down on the table. The couple will then get to turn one card over. If it’s a black ace, they’ll owe the full amount, but if it’s the ace of hearts, the waiter will give them a $20 discount. If they first turn over the ace of diamonds, they’ll then get another turn to turn over another card, earning a $10 discount if it is the ace of hearts.

a.Draw a probability model for x, the dollar value of the discount.

b.Based on the probability model, what is the expected discount for a couple?

Slide 16 - 7


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 discrete random variables as well.

The variance for a random variable is:

The standard deviation for a random variable is:

22 Var X x P x

SD X Var X


Example

Using the probability model from the previous example, find the mean (expected value) and the standard deviation in your calculator.

Slide 16 - 9


More About Means and Variances

If I add a constant to each independent random variable then I add the constant to the mean (expected value):

E(X ± c) = E(X) ± c

If I add a constant to each independent random variable then the standard deviation and variance doesn’t change:

SD(X ± c) = SD(X)

And the variance: Var(X ± c) = Var(X)


Example

The couples dining at the restaurant for Valentine’s Day discounts averaging $5.83 with a standard deviation of $8.62. There was also a coupon in the paper for $5 off any one meal (one discount per table). If every couple dining there also brings in a coupon, what will be the new mean and standard deviation of the new discount they will receive?

Slide 16 - 11


More About Means and Variances (cont.)

If I multiply each independent random variable by a constant then I multiply the mean (expected value) by the constant:

E(aX) = aE(X) If I multiply each independent random variable by

a constant then the standard deviation is multiplied by the constant:

SD (aX) = aSD (X)

And the variance: Var(aX) = a2Var(X)


Example

The couples dining at the restaurant for Valentine’s Day discounts averaging $5.83 with a standard deviation of $8.62. When two couples dine together on a single check, the restaurant doubles the discount offer -- $40 for the ace of hearts on the first card and $20 on the second. What are the mean and standard deviation of discounts for such groups of people?

Slide 16 - 13


More About Means and Variances (cont.)


Example

The couples dining at the restaurant for Valentine’s Day discounts averaging $5.83 with a standard deviation of $8.62. You go with another couple to the restaurant but they don’t like the idea of the “double discount” so they sit at the table next to you, so they can have their own discount. Does this change the expected value or the standard deviation?

Slide 16 - 15


What Can Go Wrong?

Probability models are still just models. Models can be useful, but they are not reality. Question probabilities as you would data, and

think about the assumptions behind your models.

If the model is wrong, so is everything else.


What Can Go Wrong? (cont.)

Don’t assume everything’s Normal. You must Think about whether the Normality

Assumption is justified. Watch out for variables that aren’t independent:

You can add expected values for any two random variables, but you can only add variances of independent random variables.

copyright © 2010 pearson education, inc. chapter 16 random variables

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