For a multinomial distribution where there are n trials, and three options, thus $X_1$, $X_2$, $X_3$, where all three options have an equal probability of occuring ($p_1=1/3$), what is the expected values $E(Y)$ and variance $V(Y)$ of the function Y where $Y=X_1-(X_2 + X_3)$.
I know that for the $E(X_i)=np_i $ and $V(X_i)=np_iq_i $.
But I'm lost in how to implement the function Y.
Any help greatly appreciated
For $n=1$ you have $\text E[X_i]=1/3$ and $\text{Var}(X_i)=2/9$ (Bernoulli with parameter $1/3$), so:
$\text E[Y_1]=-1/3$ and by symmetry:
$$0=\text{Var}(1)=\text{Var}(X_1+X_2+X_3)=3\text{Var}(X_i)+6\text{Cov}(X_j,X_k)$$
From where $\text{Cov}(X_j,X_k)=-1/9$, for $j\ne k$
So:
$\text{Var}(Y_1)=\text{Cov}(X_1-X_2-X_3,X_1-X_2-X_3)=3\text{Var}(X_i)-2\text{Cov}(X_j,X_k)=8/9$
Then from $n$ trials because of independence:
$$\text E[Y]=n\text E[Y_1]=-n/3$$
$$\text{Var}(Y)=n\text{Var}(Y_1)=8n/9$$