Hi, I'm confused about the equations for variance. We have: var(X+Y) = var(X) + var(Y) if X and Y independent var(nX) = (n^2)var(X) But in chapter 7 compound distributions: S= X1 + X2 + ... + XN var(S given N=n) = nvar(X) WHY NOT (n^2)var(X)? For example, say n=3. Then since all Xs are iid rvs, isn't var(S given N=3) just var(X+X+X) = var(3X) = 9var(X)?
Var(X+Y) = Var(X) + Var(Y) if X,Y independant. If considering X1,X2,X3..XN which are different observations of independant (and indentical) distributions, it's like above. Var(X1+X2+X3+...+XN) = Var(X1)+Var(X2)+Var(X3)+...Var(XN) = nVar(X) since all var are the same Note if considering Var(n*X), VAR(nX) = (n^2)Var(X) Difference here is you take one observation and multiply it by n. Note that one observation is obviously dependant on itself (above formula not valid). Say you flip a coin, X~Bin(1,0.5) and n=3. The first is where you flip 3 coins and add the number of heads together, giving you a Bin(3,0.5) The second is you flip one coin and multiply by 3, so you get 3*0 with probabilty of 0.5, and 3*1 with probability of 0.5. (Can't get 1,2)
just thinking bout this Something that perhaps adds to the confusion is that . Here you are considering the distribution of sigma( Xi ) , but when we think about the distribution of X bar the 1/n factor does the usual and gets squared so the variance goes from being n(sigma squared) to (sigma squared)/n.
