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Unformatted text preview: CS 70 Discrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 22 Variance Question: At each time step, I flip a fair coin. If it comes up Heads, I walk one step to the right; if it comes up Tails, I walk one step to the left. How far do I expect to have traveled from my starting point after n steps? Denoting a rightmove by + 1 and a leftmove by 1, we can describe the probability space here as the set of all words of length n over the alphabet { 1 } , each having equal probability 1 2 n . Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + + X n , where X i = ( + 1 if i th toss is Heads; 1 otherwise. Now obviously we have E ( X ) = 0. The easiest rigorous way to see this is to note that E ( X i ) = ( 1 2 1 ) +( 1 2 ( 1 )) = 0, so by linearity of expectation E ( X ) = n i = 1 E ( X i ) = 0. Thus after n steps, my expected position is 0! But of course this is not very informative, and is due to the fact that positive and negative deviations from 0 cancel out. What the above question is really asking is: What is the expected value of  X  , our distance from 0? Rather than consider the r.v.  X  , which is a little awkward due to the absolute value operator, we will instead look at the r.v. X 2 . Notice that this also has the effect of making all deviations from 0 positive, so it should also give a good measure of the distance traveled. However, because it is the squared distance, we will need to take a square root at the end. Lets calculate E ( X 2 ) : E ( X 2 ) = E (( X 1 + X 2 + + X n ) 2 ) = E ( n i = 1 X 2 i + i 6 = j X i X j ) = n i = 1 E ( X 2 i ) + i 6 = j E ( X i X j ) In the last line here, we used linearity of expectation. To proceed, we need to compute E ( X 2 i ) and E ( X i X j ) (for i 6 = j ). Lets consider first X 2 i . Since X i can take on only values 1, clearly X 2 i = 1 always, so E ( X 2 i ) = 1. What about E ( X i X j ) ? Since X i and X j are independent , it is the case that E ( X i X j ) = E ( X i ) E ( X j ) = 0. 1 Plugging these values into the above equation gives E ( X 2 ) = ( n 1 ) + = n . So we see that our expected squared distance from 0 is n . One interpretation of this is that we might expect to be a distance of about n away from 0 after n steps. However, we have to be careful here: we cannot 1 Two random variables X and Y are independent if the events X = a and Y = b are independent for all pairs of values a , b . If X , Y are independent, then we have E ( XY ) = E ( X ) E ( Y ) ; youll be asked to prove this on one of your homeworks. Note that E ( XY ) = E ( X ) E ( Y ) is false for general r.v.s X , Y ; as an example just look at E ( X 2 i ) in the present discussion....
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 Fall '08
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