I have a question similar to #10 here Probability: An Introduction - Google Book Search
I have a 2D symmetric random walk starting from zero. Again, D_n is the distance from the origin after n steps. I need to show that E((D_n)^2) like #10 is equal to n.
So I know that euclidean distance = sqrt((x-0)^2 + (y-0)^2), so let X_n and Y_n be the coordinates of the particle.
It seems that E(X_n)=E(Y_n)=n/2, but how do I show this?
For n = 1, E(X_1)=(.25)(0^2 + 0^2 + 1^2 + -1^2) = .5 which holds
I can do this similarly for E(X_2), E(X_3) and I get the right answers (i.e 1 and 1.5 respectively), but I'm missing the pattern. Can someone show me how to do this? Does induction work?