Transcript PowerPoint Slides for Section 4.4 - Ursinus College Student, Faculty

4.4 Mean and Variance
Mean
How do we compute the
mean of a probability
distribution? Actually,
what does that even
mean?
Let’s look at an example
on the board.
This leads us to define:
Given a discrete random
variable x with probability
distribution P(x), the
mean of the random
variable x (or P(x)) is
defined as
  xP(x)
Expected Value
Another name for the
mean in this case is
the expected value.
For a random variable
x, we define the
expected value of the
random variable,
denoted E(x), to be
E(x)=μ.
Find the expected
value of
0
1
2
3
.3
.4
.2
.1
Practice Problem
Find the expected value of the loaded die
from section 4.2.
1
2
3
4
5
6
2/15
1/6
1/6
1/6
1/6
1/5
Law of Large Numbers
We have hinted at this before, but now we
“officially” state the law of large numbers.
It states that as the number of observations
increases, the mean of the observed value
eventually approaches the mean of the
population as close as you would like.
Thus, in the long run, casinos that have the odds
in their favor will make money, for example.
A card is to be selected from an ordinary
deck of 52 cards. Suppose the casino will
pay $10 if you select an ace. If you fail,
you pay the casino a dollar. How much
does the casino expect to win per game
on average if you play this game many
times?
Suppose we are studying days of scheduled
classes missed in Colleges and Universities in
Minnesota.
We find that on average there are 1.7 days
missed due to the teacher cancelling class and
4.6 days missed due to the school cancelling
because of weather.
How many total days are missed on average in
Minnesota?
1.7+4.6= 6.3
Rules for Mean
We have illustrated the principle that if X is
random variable and Y is another random
variable, then X+Y is also a random variable.
We also have the formula μX+Y = μX+μY.
Let a, b be fixed numbers. Another formula is
μa+bX = a + bμX.
This formula says that if I multiply or add a
constant to the random variable, then the mean
changes accordingly.
Practice Problems
The random variable X has mean μX=5. If
Y=3-2X, what is μX?
The expected payoff for a game is $4.00.
If the random variable X is in terms of
dollars, convert it to cents and find the
expected payoff for that same game in
cents, justifying your answer.
Variance
There is a similar formula for
variance.
We have:
Given a discrete random
variable x with probability
distribution P(x), the variance
of the random variable x (or
P(x)) is defined as
We then define the standard
deviation as the square root of
the variance.
Notice that there is no concept
of population vs. sample.
  ( x   ) P ( x )
2
2
Example
Consider the data set
Compute the mean,
variance, and
standard deviation.
x
P(x)
1
2/15
2
10/15
3
2/15
4
1/15
Note: An easier
formula for variance is
given by
Now compute the
variance for the
above data using this
formula.
Wow, much easier!
  [x P( x)]  
2
2
2
Independence and Correlation
Two random variables X and Y are independent if
knowing that any event involving X alone did or did not
occur tells us nothing about the occurrence of any event
involving Y alone. An example is the rolling of two dice.
Dealings in a game of blackjack is an example of
dependence of events.
We may associate a correlation ρ (rho) between two
random variables. This is similar to the correlation r we
studied in chapter 2.
The correlation between two independent random
variables is 0.
Example
Scores on the Math part of the SAT have a
mean of 519 and standard deviation of 115.
Scores on the verbal part had mean 507 and
standard deviation 111.
1. What is the mean of total SAT scores?
2. What is the standard deviation of total SAT
scores?
3. Need correlation ρ = .71
Practice Problem
The amount of profit for a
major investment is uncertain,
but a probabilistic estimate
gives the following distribution
(in millions of dollars).
Find the mean profit and the
standard deviation of the profit.
The investment firm owes its
source of capital a fee of
$200,000 plus 10% of the
profits. The firm thus retains
Y=.9X-.2 from the investment.
Find the mean and standard
deviation of Y.
Profit
1
1.5 2
4
10
Probability
.4
.2.
.1
.1
.2