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The Mean and Variance of a
Random Variable
•
Two important quantities that
describe the behavior of a random
variable and PMF.
•
The mean is also equivalently known
as the Expected Value.
•
We can denote the mean/expected
value or expectation of a random
variable X using
–
E[X] or
zX
or often just I
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Example 1
•
Two loaded dice (X, and Y) have the
X
1
2
3
4
5
6
PX(x)
0.01
0.05
0.1
0.2
0.3
0.34
PY(x)
0.34
0.3
0.2
0.1
0.05
0.01
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Example
•
Find the mean of the Bernoulli
Distribution.
p
•
Note that when a PMF is written in
generic form with parameters, the
mean
is a
function of the parameters
•
Find the mean of the Geometric
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Important Properties of
Suppose X is a random variable, and
a
and
b
are constants.
Suppose we
define Y as follows:
Y = aX + b
Then Y is a random variable, but with a
Suppose X was the outcome of a die,
and Y = 2X+10.
Then the PMF of Y is:
y
12
14
16
18
20
22
PY(y)
1/6
1/6
1/6
1/6
1/6
1/6
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Property 1
Suppose we create a new random
variable Y from X as Y = aX+b.
Given
the mean of X is E(X), what is the
mean of Y?
E[Y] = E[aX+b] = aE[X]+b
Proof:
•
PY(y) = P(Y=y) = P(Y=ax+b) =
PX(x)
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Property 2
•
Let g() be any function, and X be a
random variable.
Let Y = g(X), thus
Y is a random variable as well.
Then:
•
E[g(X)] =
g(x) PX(x)
However the following is
NOT
CORRECT
•
E[ g(X)] = g( E[X] )
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Property 3
Suppose we have two random
variables X and Y.
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This note was uploaded on 01/04/2010 for the course STATS 1100 taught by Professor Rodriguez during the Spring '08 term at Pittsburgh.
 Spring '08
 Rodriguez
 Statistics, Probability, Variance

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