The expected length of code for one encoded symbol is
where is the probability of picking the letter
, and
is the length of code needed to encode
.
is given to us, and we have
so that we expect a contribution of
bits to the code per encoded letter. For a string of length , we would then expect
.
By definition of variance, we have
For a string consisting of one letter, we have
so that the variance for the length such a string is
"squared" bits per encoded letter. For a string of length , we would get
.