Question

from a box of fruit containing 8 oranges and 1 apple a random sample of 2 pieces of fruit has been selected without replacement. Let X be the number of oranges and Y be the number of apples in the sample. What will the correlation of X and Y be? (Hint the variance of X and Y same. Covariance is just negative of these variances. )

Answer 1

X can take values 1,2 and Y can take values 0,1

(X,Y) can take values (1,1) or (2,0)

P(X = 1, Y = 1) = 8C1*1C1/9C2 = 2/9

P(X = 2, Y = 0) = 8C2/9C2 = 7/9

E(XY) = 1*1*2/9 + 2*0*7/9 = 2/9

Marginal distribution of X

P(X = 1) = 2/9, P(X = 2) = 7/9

E(X) = 1*2/9 + 2*7/9 = 16/9

Var(X) = = 14/81

Marginal distribution of Y

P(Y = 0) = 7/9

P(Y = 1) = 2/9

E(Y) = 2/9, Var(Y) = 14/81

Covariance of X and Y, Cov(X,Y) = E(XY) - E(X)E(Y)

= -14/81

Correlation of X and Y =

= -1

Thus, the correlation of X and Y is -1