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40 percent of the students in a large population are freshmen, 30 percent are sophomores, 20 percent are juniors, and 10 percent are seniors.X1, X2, X3, X4 denote, respectively, the numbers of freshmen, sophomores, juniors, and seniors that are obtained .We need to determine

a. 蟻(Xi, Xj ) for each pair of values i and j (i< j ).

b. For what values of i and j (i<j )is 蟻(Xi, Xj ) most negative.

c. For what values ofi and j (i<j ) is 蟻(Xi, Xj ) closest to 0.

(a) The discrete random vector \({X_i}\) has multinomial distribution with parameters n and \({p_i}\) where \({p_i}\)=0.4,0.3,0.2,0.1 for i=1,2,3,4 respectively.

\(E\left( {{X_i}} \right) = n{p_i},{\rm{Var}}\left( {{X_i}} \right) = n{p_i}\left( {1 - {p_i}} \right),{\rm{Cov}}\left( {{X_i},{X_j}} \right) = - n{p_i}{p_j}\)

From definition for i<j,

\(\begin{aligned}{}\rho \left( {{X_i},{X_j}} \right) &= \frac{{{\rm{cov}}\left( {{X_i},{X_j}} \right)}}{{\sqrt {{\rm{Var}}\left( {{X_i}} \right)} \sqrt {{\rm{Var}}\left( {{X_j}} \right)} }}\\ &= \frac{{ - n{p_i}{p_j}}}{{\sqrt {n{p_i}\left( {1 - {p_i}} \right)} \sqrt {n{p_j}\left( {1 - {p_j}} \right)} }}\\ &= \frac{{ - {p_i}{p_j}}}{{\sqrt {{p_i}\left( {1 - {p_i}} \right)} \sqrt {{p_j}\left( {1 - {p_j}} \right)} }}\end{aligned}\)

Putting the value of p where \({p_i}\)=0.4,0.3,0.2,0.1 for i=1,2,3,4 respectively.

we get the following pairs

b. From the table , it can be stated that i=1, j=2,蟻 is most negative.

c. From the table, it can be stated that i=3,j=4,蟻 is the closest to 0.