Transcript Slide 1

Chapter 16
Random Variables
Copyright © 2009 Pearson Education, Inc.
Objectives:
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The student will be able to:
 Recognize random variables.
 Find the probability model for a discrete
random variable.
 Find and interpret in context the mean
(expected value) and the standard deviation of
a random variable.
Copyright © 2009 Pearson Education, Inc.
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Random Variables
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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 a lower case letter, in this case x.
 Examples:
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Let X = the roll of a die
Let X = five times the roll of a die
Let X = number of rolls of a die until you roll a 6
Let X= the sum of 3 rolls of a die
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We might ask what is the P(X=5) for example
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Random Variables
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There are two types of random variables:
 Discrete random variables can take one of a
finite number of distinct outcomes.
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Example: Number of credit hours
Continuous random variables can take any
numeric value within a range of values.
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Example: Cost of books this term
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Probability Models for Random Variables
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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.
 We’ll make a table x, f(x), P(x)
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Let X = the roll of a die
Let X= the sum of 2 rolls of a die
Let X = five times the roll of a die
Let X = number of rolls of a die until you roll a 6
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Expected Value: Center
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We can describe these models using the
methods we’ve seen before. We can look at the
histogram for discrete random variables or a
graph for a continuous random variable
We are particularly interested in the value we
expect a random variable to take on. We’ll
denote this using μ (for population mean) or E(X)
for expected value.
 Lets look at the probability model for X = the
sum of rolling two dice. We need a table of x,
f(x), and P(x),
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Expected Value: Center (cont.)
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The expected value of a (discrete) random
variable can be found by summing the products
of each possible value and the probability that it
occurs:
  E  X    x  P  x
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Note: Be sure that every possible outcome is
included in the sum and verify that you have a
valid probability model to start with.
Compute E(X) for X = the sum of rolling two dice.
Copyright © 2009 Pearson Education, Inc.
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Example (from text)
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A restaurant gives every couple an opportunity to
win a discount. The waiter brings 4 aces to the
table. If the couple picks a black ace, they owe
the full amount. If they pick the ace of hearts,
they’ll get a $20 discount. If they pick the ace of
diamonds, they get to pick one of the remaining
cards and if it’s the ace of hearts, they’ll get a $10
discount.
What is the expected discount for a couple?
 We need to make a table: Outcome, x, P(X=x)
 E(x) = Σ x*P(x)
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First Center, Now Spread…
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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:
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  Var  X     x     P  x 
The standard deviation for a random variable is:
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2
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2
  SD  X   Var  X 
Compute the standard deviation for the sum of
rolling two dice.
Do the same for the restaurant example
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Using the TI-83
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Much of what you have to do for discrete random data is very much
like what we did to get the descriptive statistics for data involving
frequencies only here we will use decimals, probabilities.
Example:
 A recent study involving households and the number of children in
the household showed that 18% of the households had no
children, 39% had one, 24% had two, 14% had three, 4% has
four and 1% had five. Find the expected value, mean number of
children per household, and the standard deviation of this
distribution.
Place the values of the variable in L1: 0,1,2,3,4,5 and the associated
probabilities in L2: .18, .39, .24, .14, .04 and .01.
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Using the TI-83
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Then do Stat -> Calc -> 1-var stat(L1 , L2)
 The mean is 1.5 which is also the sum…. the calculator is doing
what we would do if we used the formula… multiply each value by
its corresponding probability and then add the separate products.
 The standard deviation is 1.1180. Notice the n=1. That just means
that the individual probabilities added up to one which they should
if this is a proper distribution.
If you need to see the probability histogram, think about the classes
and the boundaries: -.5 to .5; .5 to 1.5 etc. The y values are the
probabilities which go no lower than 0 and no higher than .39. So
here is a useable window:
 xmin
-.5 xmax 5.5 xscl 1
 ymin
0 ymax .39 or a little higher if you don't want the bars to
go to the top of the screen
 leave yscl and xres alone. Hit the graph button.
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Practice
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4) A day trader buys an option on a stock that will
return $100 profit if it goes up today and loses $200 if
it goes down. If the trader thinks there is a 75%
chance that the stock will go up, what is his expected
value of the option?
6,14) You roll a die. If it comes up a 6, you win $100.
If not, you get to roll again. If you get a 6 the second
time you win $50, if not, you lose.
 Create a probability model for this game
 Find the expected amount you’ll win
 How much would you be willing to pay to play this
game?
 Find the standard deviation of the amount you
might win
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Copyright © 2009 Pearson Education, Inc.
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24) Your company bids for two contracts. You
believe the probability that you get contract #1 is 0.8.
If you get contract #1, the probability that you get
contract #2 is 0.2, and if you do not get contract #1,
the probability that you get #2 will be 0.3.
a)
Are the two contracts independent? Explain
b)
Find the probability that you get both contracts
c)
Find the probability that you get no contract
d)
Let X be the number of contracts you get. Find
the probability model for X
e)
Find the expected value and standard deviation
of X
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Stop!
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The rest of the slides are FYI… and will not be on
the exam
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More About Means and Variances
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Adding or subtracting a constant from data shifts
the mean but doesn’t change the variance or
standard deviation:
E(X ± c) = E(X) ± c
Var(X ± c) = Var(X)
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Example: Consider everyone in a company
receiving a $5000 increase in salary.
Suppose we adjust both our dice to make the
numbers run 2 through 7 instead of 1 through
6, how would this affect the mean and
standard deviation of the sum of rolling both
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More About Means and Variances (cont.)
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In general, multiplying each value of a random
variable by a constant multiplies the mean by that
constant and the variance by the square of the
constant:
E(aX) = aE(X)
Var(aX) = a2Var(X)
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Example: Consider everyone in a company
receiving a 10% increase in salary.
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More About Means and Variances (cont.)
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In general,
 The mean of the sum of two random variables
is the sum of the means.
 The mean of the difference of two random
variables is the difference of the means.
E(X ± Y) = E(X) ± E(Y)
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If the random variables are independent, the
variance of their sum or difference is always
the sum of the variances.
Var(X ± Y) = Var(X) + Var(Y)
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Combining Random Variables (The Bad News)
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It would be nice if we could go directly from
models of each random variable to a model for
their sum.
But, the probability model for the sum of two
random variables is not necessarily the same as
the model we started with even when the
variables are independent.
Thus, even though expected values may add, the
probability model itself is different.
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Continuous Random Variables
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Random variables that can take on any value in a
range of values are called continuous random
variables.
Continuous random variables have means
(expected 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 these parameters.
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Combining Random Variables (The Good News)
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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 Normal models, so does their sum
or difference.
This fact will let us apply our knowledge of
Normal probabilities to questions about the sum
or difference of independent random variables.
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