### 4.2    Expectation, Variance, Moments, Moment Generating Function

Consider:  in discrete case, the expected value of a random variable X is defined as
i.e., it gives the weighted average of all possible values of X
• can’t quite do this in continuous case, since  P(X=x) = 0  for every x!
• still, want  E(X) = weighted average of all possible x values

Try instead:  approximate the continuous distribution by a discrete one, as follows:
Suppose possible values for X lie within some interval [a,b].

• break the range of possible values into n subintervals of equal width; let the points of subdivision be

• a = x0, x1, x2, ... , xn = b;  the width of each subinterval will be  Dx =
• from the definition of the density function, the probability that X lies between xi and xi+1  = area under density curve from xi to xi+1, i.e.,

• as long as Dx is small.
• approximate X by a discrete random variable which takes on the values x0, x1,  ... , xn-1  -  i.e., if the value of X lies between xi to xi+1, round it down to xi . Then the expected value for this "discretized" approximation to X is given by the sum of the products of the x values and the probabilities from above:

• get a closer approximation to X by using more closely spaced points; should get "true" value for the expected value of X in the limit as the number of subintervals approaches infinity; and thus should have

• by the definition of the definite integral.
We use the above to motivate the definition of the expected value of a continuous random variable (extending it to the case wgere the possible values lie in the infinite range from - to ):

Def:   If X is a continuous random variable, then its expected value is defined as

• as before, we also call this the mean of X, denoted m

•
ex:
Let X be the weight of cereal in a randomly selected box, as in the example from the previous section, and suppose it has the density function
Then the expected value of X is
(note that  f(x) = 0  for x < 14 and x > 15)
=     =  14.33
Note that the value of the mean is less than the midpoint of the interval [14, 15] of possible values; this follows because the density function is higher on the left end of the interval, indicating that X is more likely to lie close to 14 than close to 15.

In general,  E(X) gives the “balance point” of the density function: if the region between the density curve and the x-axis were cut out of a piece of wood, the location of the mean would be the point on which the piece would balance.

For any function H(X) of X, we define the expected value of H as

Thus the moments are

The variance is
We have the same properties for expectation as before:
1. E(cX) = c E(X)
2. E(X+Y)  =  E (X) + E (Y)
As before, these give an alternate formula for computing the variance:
var(X) = E(X2) - E(X)2

ex:

For the above "cereal density",
=     =
=    205.5
So
var(X)  =  E(X2) - E(X)2  =  205.5  -  (14.33)2   =   .0565
s  =  .24

Note:  the standard deviation measures the expected deviation from the mean, as before; thus it measures the spread of the density function, i.e., how widely spread the values tend to fall from the location of the mean.

The moment generating function is again defined as

and is used as before to find the values of the moments by differentiation:

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