Lecture 9/ EXPECTATIONS AND MOMENTS 2017 2018 /

EXPECTATIONS AND MOMENTS
An extremely useful concept in problems involving random variables or distributions is that of expectation. The subsections of this section give definitions and results regarding expectations.
Mean
Definition 10 Mean Let X be a random variable. The mean of X,
denoted by or E[X], is defined by: E[X]=?_(-?)^???xf(x)dx? (i)
if X is discrete with mass points ?_(?x)??xf(x)? (ii)
E[X] is defined to be the indicated series provided that the series is
absolutely convergent; otherwise, we say that the mean does not exist. And in (i), E[X] is defined to be the indicated integral if the integral exists; otherwise, we say that the mean does not exist. E[X] is the center of gravity (or centroid) of the unit mass that is determined by the density function of X. So the mean of X is a measure of where the values of the random variable X are" centered."
Expectattion of a Function of a Random Variable. Consider a r.v.X
with p.d.f. (p~m.f.)){f) and distribution function F(x). If g (.) is a function such
that g (X) is a r.v. an)) E [g (X)] exists (i.e., is defined), then

E [g (X)]=?_(-?)^???g (x)f(x)dx ?
=?_(?x)??g (x)f(x) ?
If X and Y have a joint p.d.f..f(x, y) and Z=h (x,y) is a random variable for some
_
E (Z) =?_(-?)^???h( x ,y )f ( x ,y )dxdy? if X, Y are continuous
or
=?_(?y)??_(?x)??h( x ,y )f ( x ,y ) ? if X, Y are discrete, and
E (X^r )=?_(-?)^???x^r f(x)dx? or
=?_(?x)??x^r f(x)?