Transcript The Mean of a Random Variable

The Mean of a Discrete RV
• The mean of a RV is the average value the RV takes
over the long-run.
– The mean of a RV is analogous to the mean of a large
population.
– The mean of a RV is different than a sample mean, which is
the average of a sample of size n taken from a population.
• The mean of the RV X is denoted by mX.
• The mean is also called the expected value, denoted
E(X).
The Mean of a Discrete RV
• The mean of a discrete RV X that takes k different
values with probability pi for the ith value, the mean is:
m X  E ( X )  x1 p1  x2 p2    xk pk

k
x p
i 1
i
i
• The mean is the sum of the values of the RV, weighted
by the probabilities of the values.
The Variance of a Discrete RV
• The variance of a RV is a measure of the spread in the
probability distribution of the RV about the mean.
– The variance of a RV is analogous to the variance of a large
population.
– The variance of a RV is different than the sample variance.
2
s
• The variance of a RV X is denoted by X.
• The standard deviation of X is the square root of the
variance, denoted by sX.
The Variance of a Discrete RV
• For a discrete RV X that takes k different values with
probability pi for the ith value, the variance is:
s  x1  m X  p1  x2  m X  p2    xk  m X  pk
2
2
X

k
2
 x  m 
i 1
2
i
X
2
pi
• The variance is a sum of the squared distances
between the values of the RV and its mean, weighted
by the probabilities of the values.
Mean & Variance of
Continuous RVs
• We can find the mean and variance of a
continuous random variable, but we need to
use calculus techniques to do so.
• Beyond the scope of MATH 106.
Mean & Variance of a
Linear Function of a RV
• Let Y = a + bX, where X is a RV with mean
mX and variance s X2 .
• The mean of Y is:
mY  mabX  a  bm X
• The variance of Y is:
s s
2
Y
2
a bX
b s
2
2
X
Sums of Independent RVs
• Let X and Y be independent random
variables. Then
m X Y  m X  m Y
s
2
X Y
2
X
 s s
2
Y
s
2
X Y
 s s
2
Y
2
X
Sums of Dependent RVs
• Let X and Y be dependent random variables.
Then
m X Y  m X  m Y
s
2
X Y
 s  s  2r s Xs Y
s
2
X Y
 s  s  2r s Xs Y
2
X
2
X
2
Y
2
Y
where r is the correlation between the
random variables X and Y.
The Law of Large Numbers
• As the sample size n (from a population
with finite mean m) increases without
bound, the sample mean x approaches m.