Discrete Random Variables - Mean and Standard Deviation

1. Nov 5, 2013

dyno0919

1. The problem statement, all variables and given/known data
There are a set number of marbles in a bag; the marbles consist of two colors. We are given the mean number of marbles of color 1 in the bag, as well as color 1's standard deviation. We are then asked to find the mean and standard deviation of color 2.

2. Relevant equations
How do we find the standard deviation of one variable given the standard deviation of another variable?

3. The attempt at a solution
I believe the mean of color 2 would just be (# of marbles in bag)-(mean # of color 1 marbles), but I may be wrong. We haven't learned anything about covariance, and after a sufficient amount of googling that seems like the only way to connect two variances (and thereby two standard deviations).

If we assign X=# of marbles of color 1 and Y=# of marbles of color 2, then it makes sense that E[X+Y]=E[X]+E[Y], or E[Y]=E[X+Y]-E[X]. But then, Var(X+Y) would be 0 because the number of marbles in the bag never changes? Then if Var(X+Y)=Var(X)+Var(Y), we end up with a negative variance for Y... It is at this point that I haven't got a clue how to proceed.

2. Nov 5, 2013

Ray Vickson

You cannot write Var(X+Y) = Var(X) + Var(Y) because X and Y are correlated. Instead, use the standard result that Var(Y) = E(Y^2) - (EY)^2, and re-express everything in terms of X.