Nasty.

That would be a partial answer, too, as many camera use math in addition
to amplification to increase virtual sensitivity.

A RAW converter (or the in-camera JPEG engine) can hide shadow noise a
bit by darkening the shadows, but noise exists at all levels.  Most of
the random noise is blackframe noise, which would be there even if the
lenscap were on.  The signal simply adds to it in the sensor.

High ISO is usually associated with noisey images, but it is important
to understand why.  If you leave a camera on a tripod, point it at the
same scene, in an auto-exposure mode, and change the ISO to take
pictures, the reason that the high-ISO images are noisier is because the
metering system of the camera forces a lower sensor exposure for the
higher ISOs, since the camera is then using a fraction of the sensor's
dynamic range to create the digitized image.  As the camera changes the
DR used in the sensor, the amount of noise in the sensor remains the
same, so the signal-to-noise ration is lower, and you get a noisier
image.

It is not true, however, that a higher ISO means more noise relative to
a certain signal level in the sensor; in fact, just the opposite is true
for many cameras.  The image noise, relative to signal in the sensor (as
contrasted to signal in the range used by the ISO) is actually lower at
higher ISOs, and part of the reason for this is that the same number of
digitized levels (or very similar) are used at various ISOs, so the same
sensor signal gets coarser digitization at the lower ISOs.  With some
cameras, noise increases at a slower rate than ISO increase by a
considerable amount.  The Canon 20D has total blackframe noise that has
standard deviations that increase like this:

 ISO    Std Dev

 100        2.1
 200        2.2
 400        2.4
 800        3.2
1600        4.7

Those are based on RAW values, so they are relative to the fraction of
the DR in the sensor used by the ISO.  Relative to absolute sensor
signal, using ISO 100 as a standard, we get:

 ISO    Std Dev

 100        2.1
 200        1.1
 400        0.6
 800        0.4
1600        0.3

So, if you under-expose an ISO 100 image at -2 EC, you will have 7x as
much noise (relative to signal) as if you had shot it at ISO 1600 with
+2 EC (this assumes that the scene is of sufficiently low contrast that
+2 EC does not clip desired highlights).  This is important to
understand, as many people seem to blindly assume that they will always
get less noise by using lower ISOs.  With this particular camera (Canon
20D), there really is very little reason to go below ISO 400, if you are
on a limited photon budget.  ISO 100 is actually a little crippled on
many cameras, as the RAW data may be clipped low or distorted to
accomodate sensor saturation, causing color issues in extreme
highlights.  My 20D never goes below ISO 200 anymore, unless I really
need a longer shutter speed or a wider aperture, and don't want to blow
the highlights.

Here is a visual demonstration.  Both crops are shot 10 seconds apart
(same light), on a tripod, with the same shutter speed and f-stop in
manual exposure mode, except that one is ISO 100, and the other is ISO
1600.  The entire image was exposed at ISO 1600 for +2 EC, and the crops
are from a much darker area, are pushed to ISO 6400 in the 10D image,
and ISO 10,000 in the ISO 1600 image.  The images are the original RAW
data, with blackpoint (128) subtracted first, color channels
interpolated to fill a full 8.2MP for each color, scaled for the push,
white-balanced, and a 2.2 gamma applied.  The image colors are dull, but
that's the way RAW color is; you can see, however, how saturated the
color of the noise is in the ISO 100 image:

http://www.pbase.com/jps_photo/image/57149515/original

So, one can conclude that the higher ISO does a much better job of
capturing the DR of the sensor to which it is assigned.

Since the camera does not know noise from signal, noise gets amplified
too.  Astronomers get around this by making many long exposures adding
up signals above a threshold and deleting signals below that threshold,
but this technique is of very limited value in short duration, single
exposure photography.

Cheers,
Alan