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BIOINF 2118
N 05 - Expectation and Variance
Page 1 of 4
“Expectation” is a “measure of central tendency”, one kind of “average”.
Expectation = “mean” .
Expectation = value of a bet, or a gamble (Rev.Thomas Bayes).
Expectation = balance point.
Other “averages” include:
- the median (the 50% quantile,
- the mode (the most common value).
Definition of expectation:
The expected value of a random variable X
= E[X] or E(X) or EX
= the mean of the distribution
= the mean of X.
“Units” = x-thingies
).
BIOINF 2118
N 05 - Expectation and Variance
Page 2 of 4
For a discrete random variable (RV):
The expected value is only defined if
. (“Absolute convergence “)
For a continuous RV:
where f is the density (p.d.f.) of X. The expected value is only defined if
(“Absolute convergence “)
(How can the mean NOT exist?
The Cauchy distribution …)
The expectation of a function r( ) is
or
Let X and Y have a joint distribution with pmf or pdf f. If r is a function of X and Y, then
the expected value of r(X,Y) is defined by
or
depending on whether X and Y are discrete or continuous. Absolute convergence is
still required for the expected value to be defined.
BIOINF 2118
N 05 - Expectation and Variance
Page 3 of 4
The Laws of Large Numbers
What’s so special about expectation?
The distribution of the sample mean
converges to the point distribution
with a p.m.f. equal to 1 on E(X ) and zero everywhere else,
X1
X2
X3
Average ----------------------------------------------->
The Variance of a RV
The variance, or 2nd central moment, of a RV X is defined as
The variance measures the spread of a distribution.
“Units” = square-x-thingies
Example: If
, then the mean of X is
- If X 1,...,X n are i.i.d. with mean
and variance
and the variance is
.
,
then what are mean and variance of X ?
- What are the mean and variance of binomial? Bernoulli? uniform? Poisson?
The Standard Deviation is the square root of the variance. “Units” = x-thingies.
The Coefficient of Variation is Standard Deviation / Mean. Scale-free!!
(The CV only makes sense if X is non-negative.)
BIOINF 2118
N 05 - Expectation and Variance
Page 4 of 4
Higher moments
The kth central moment, of a RV X is defined as
Any distribution that is symmetric around the mean has all odd central moments = 0.
The skewness measures the lop-sidedness of a distribution.
.
Notice that it is scale-free.
The kurtosis measures the lumpiness of a distribution.
.
Notice that it is also scale-free. If X is normal, kurtosis = 0.
Covariance
For two RV’s X and Y, the covariance between them is
.
Correlation
To get scale-free measure of association, we define the correlation,
.
Scale-free! (“Dimensionless”)