Suppose that – in any given time period – a certain stock is equally likely to go up 1 unit or down 1unit, and that the outcomes of different periods are independent. Let X be the amount the stock
goes up (either 1 or -1) in the first period, and let Y be the cumulative amount it goes up in the first three periods. Find the correlation between
X and Y. [Hint: Let Xi be the amount the stock goes up in the i-th time period so Y = X1+X2+X3]

Question

Suppose that – in any given time period – a certain stock is equally likely to go up 1 unit or down 1unit, and that the outcomes of different periods are independent. Let X be the amount the stock
goes up (either 1 or -1) in the first period, and let Y be the cumulative amount it goes up in the first three periods. Find the correlation between
X and Y. [Hint: Let Xi be the amount the stock goes up in the i-th time period so Y = X1+X2+X3]

anonymous · Answer

So, in any given time period, we get either U or D with each 1/2 probability.  If that is the scenario, in 3 times periods, we could have UUU, UUD, UDU, UDD, DUU, DUD, DDU, DDD with each of these at 1/8 probability.  For example, UUU is 3+.  So, 3+ is 1/8, 1+ is 3/8, 1- is 3/8 and 3- is 1/8.

anonymous · Answer

Oh ok, I see now. So do I now use this equation now? 
$$P(X,Y)=Cov(X,Y)/\sqrt{Var(X)Var(Y)}$$

anonymous · Answer

not sure.