Transcript Random Variables

Random Variables
Chapter 16
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 a lower case letter,
in this case x.
Slide 162
Expected Value: Center (cont.)

There are two types of random variables:
 Discrete
random variables can take one of a
finite 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
Slide 163
Expected Value: Center (cont.)

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.
Slide 164
Example

You pick one card from a deck of cards. If you pick a face card you win
$15. If you pick an ace you win $25, and if you pick the ace of hearts
you win $65. For any other card you win nothing.

Create a probability model for the amount you win at this game.
Amount
won
P(Amount
won)
$0
$15
$25
$65
36/52
12/52
3/52
1/52
Expected Value: Center (cont.)

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:
  E  X    x  P  X  x

Note: Be sure that every possible outcome is included in the
sum and verify that you have a valid probability model to
start with.

Find the expected value of the card example
Slide 166
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:
  Var  X     x     P  X  x 
2
 The
2
standard deviation for a random variable is:
  SD  X   Var  X 
Slide 167
More About Means and Variances

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)
 Example:
What happens if we add $100 to each
of the winning amounts of the card game.
Slide 168
More About Means and Variances (cont.)

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)
 Example:
Consider everyone in a company
receiving a 10% increase in salary.
Slide 169
More About Means and Variances (cont.)

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)

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)
Slide 1610
example

Miguel buys a large bottle and a small bottle of
juice. The amount of juice that the manufacturer
puts in the large bottle is a random variable with
a mean of 1024 ml and a standard deviation of 12
ml. The amount of juice that the manufacturer
puts in the small bottle is a random variable with
a mean of 508 ml and a standard deviation of 4
ml. Find the mean and standard deviation of the
total amount of juice in the two bottles.
Example
 Suppose
that in one town, 50 year old men
have a mean weight of 177 lb. with a
standard deviation of 17 lb. 30 year old
men have a mean weight of 158 lb. with a
standard deviation of 12 lb. How much
heavier do you expect a 50 year old man to
be than a 30 year old man and what is the
standard deviation of this difference?
Example
A
national study found that the average
family spent $422 a month on groceries,
with a standard deviation of $84. The
average amount spent on housing (rent or
mortgage) was $1120 a month, with
standard deviation $212. What is mean and
standard deviation of the total?
Example

In the 4 × 100 relay event, each of four runners runs 100
meters. A college team is preparing for a competition. The
means and standard deviations of the times (in seconds)
of their four runners are as shown in the table What are
the mean and standard deviation of the relay team's total
time in this event? Assume that the runners' performances
are independent.
Runner
1
Mean 12.17
SD
0.10
2
12.24
0.15
3
11.94
0.08
4
11.79
0.08
Total
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 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.
Slide 1615
Example

Sue buys a large packet of rice. The amount of
rice that the manufacturer puts in the packet is a
random variable with a mean of 1016 g and a
standard deviation of 8 g. The amount of rice that
Sue uses in a week has a mean of 210 g and a
standard deviation of 40 g. If the weight of the
rice remaining in the packet after a week can be
described by a normal model, what's the
probability that the packet still contains more
than 895.7 g after a week?