## Definition and Examples of Expected Value |

Suppose that *X* is a random
variable for an experiment,
taking values in a subset *S* of **R**.** **If *X*
has a discrete distribution with
density function *f *then the *expected value* of *X*
is defined by

If *X* has a continuous
distribution with density function *f* then the *expected
value* of *X* is defined by

Finally suppose that *X* has a mixed distribution. Specifically,
assume that *S* is the union of disjoint sets *D*
and *C* such that the conditional distribution of *X*
given *X* in *D* is discrete with density *f*_{D}
and the conditional distribution of *X* given *X*
in *C* is continuous with density *f*_{C}
. Let *p* = *P*(*X* in *D*). The *expected
value* of *X* is

In any case, the expected value of *X* may not exist
because the sum or the integral may not converge. However, unless
otherwise noted, whenever we write an expected value, we are
assuming that it exists.

The mean is the center of the probability
distribution of *X.* Indeed, if we think of the distribution
as a mass distribution, then the mean is the center of mass as
defined in physics.

To understand expected value in a probabilistic way, suppose that we create a new, compound experiment by repeating the basic experiment over and over again. This gives a sequence of independent random variables,

X_{1},X_{2},X_{3}...

each with the same distribution as *X*. In statistical
terms, we are sampling from
the distribution of *X*. The average value, or sample mean,
after *n* runs is

Then the *average* value will converge to the *expected
*value as *n* increases. The precise statement of this is
the law of large numbers.

** 1.** Let *I* be an indicator
random variable (that is, a variable that takes only the values 0
and 1). Show that

E(I) =P(I= 1)

** 2.** Suppose that *X* is uniformly
distributed on a finite subset *S* of **R**.
Show that *E*(*X*) is the arithmetic average of the
numbers in *S.*

** 3.** Suppose that *X* is uniformly
distributed on an interval (*a, b*) of **R.** Show that
the mean is the midpoint of the interval:

E(X) = (a+b) / 2

** 4. **Find the mean of the random
variable *X *that has probability density function *f *given
by

f(x) =x^{2}/ 3 for -1 <x< 2

** 5.** Suppose that *X* has the power
distribution with parameter *a *> 1, which has
density

f(x) = (a- 1)x^{-a}forx> 1

Show that

** 5.** Suppose that *X* has the *Cauchy**
distribution* with density function

Show that the expected value of *X* does not exist.

## Expected Value |