Charlie's utility function is U(A, B) = AB, where A and B are the numbers of apples and bananas, respectively, that he consumes.
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you cannot show too much "work"
basically, you remove what is common to all of the factors, and then put brackets, as it will be multiplied back in, remember that when you multiply exponents with the same base, its same as adding them, so subtract to remove...
you can seperate two of the variables , then factor, then subtract the last one from those two, because it cannot be factored out , as in part2 #2
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
.