As we already mentioned in the previous
sections, in order to estimate the class-conditional density *p(x|X)*
an assumption is usually made that *x|X* is distributed normally with
unknown parameters. In the framework of this tutorial we'll make another
simplifying assumption that the covariance matrix is known.

In this section we'll focus on the case where the feature vectors are one-dimensional. That is, the only parameter in the parameter vector is the mean . Recall from the approach section that the a priori density of the parameter vector is known. In our case, we'll assume that

that is the mean
is distributed normally with the parameters and *.*

Moreover, the parameter vector
completely defines the probability density function *p(x)* which means

where variance is known and mean is the parameter.

Let's now look how to compute
*p(
|X). *
It holds that

where *n* is the number
of samples and *c _{1} *is a constant independent of .
Since given ,

Therefore,

Let's pay attention that

Therefore,

It's easy to verify that *p( |X)*
is also distributed normally, that is

where the parameters
and
depend on the number of samples *n*. It holds that

where

Now we are ready to compute the desired class-conditional density function:

Therefore,
*p(x|X)*
which is equal to the class-conditional density function
*p(x|w _{k},X_{k})
*is normally distributed:

Let's now look at an example illustrating the formulas we developed.

Suppose *p(x| )~N(
,
4)* where *p(
)~N(0,1)*.

The figure above contains *50* samples.

Let's see how *p( |X)
*gets
changed* *as number of samples increases from *10* to
*50*:

Since approaches
zero as *n *goes to infinity, the density function gets sharper as
the number of samples increases.

Click here to play with an applet illustrating the theory discussed above.