Question

Covariance:

Suppose 10 balls are randomly distributed into 4 urns. Let Xj be the number of balls that fall into the jth urn, and let Ti be the indicator variable that the ith ball falls into jth urn, with E {1, . . . , 10} and j є {1, . . . , 4). ·P(X, = 0) =?

Answer 1

$T_{i,j}=1~with~probability=\frac{4^9}{4^{10}}=\frac{1}{4}\\ =0~with~probability=1-\frac{1}{4}\\ E(T_{i,j})=P(T_{i,j}=1)=\frac{1}{4}=0.25\\ Var(T_{i,j})=E(T_{i,j}^2)-E^2(T_{i,j})=P(T_{i,j}=1)-\frac{1}{4^2}=\frac{3}{4^2}=3/16=0.1875 \\ P(X_j=0)=\frac{3^{10}}{4^{10}}=0.0563\\ P(X_{j}=1)=\binom{10}{1}\frac{3^{9}}{4^{10}}=\binom{10}{1}\left(\frac{1}{4} \right )\left(\frac{3}{4} \right )^9=0.1877\\ E(X_j)=\sum_{i=1}^{10}E(T_{i,j})=\frac{10}{4}=2.5\\ Var(X_j)=\sum_{i=1}^{10}Var(T_{i,j}),~since~T_{i,j}'s~are~independent\\ =30/16=1.875\\$