Question

From a box of fruit containing 75 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 covariance of X and Y be?

Answer 1

Answer:

Given that,

From a box of fruit containing 75 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 covariance of X and Y be:

Therefore, X could be 1, 2

Y could be 0,1

We have 75 oranges, 1 apple

Then,

P(X=1, Y=1):

=0.0263

P(X=2, Y=0):

=0.9737

X		Y
0	1	Total
1	0	0.0263	=0.0263
2	0.9737	0	=0.9737
Total	=0.9737	=0.0263	=1

Therefore,

PMF of X is,

X	1	2
P(X)	0.0263	0.9737

To find E(X):

=1 0.0263+2 0.9737

=0.0263+1.9474

=1.9737

Therefore, E(X)=1.9737

PMF of Y is,

Y	0	1
P(Y)	0.9737	0.0263

To find E(Y):

=0 0.9737+1 0.0263

=0.0263

Therefore, E(Y)=0.0263

To find E(XY):

= 1 1 0.0263+2 0 0.9737

=0.0263

Therefore, E(XY)=0.0263

Then,

Covariance of X and Y:

=0.0263-1.9737 0.0263

=0.0263-0.0519

=-0.0256

Therefore, Cov(X,Y) is -0.0256.