Question

A bag contains 4 red and 5 green balls. Let X denotes number of red and Y denotes number of green balls in 2 drawn from the bag by random. Find joint probability distribution, compute E(x), E(Y) and correlation coefficient.

Answer 1

Since "x" denotes the Red Balls and "y" denotes the green balls.

We randomly draw 2 balls.

Now, we make the probability distribution table of X and Y

Remember for joint probability P(X,Y) > or < 2 =0

Only X+Y =2 will give a value for probability.

WHEN RED BALLS IS DRAWN WE HAVE PROBABILITY DISTRIBUTION AS FOLLOWS. NO. OF RED BALLS DRAWN WILL BE 0,1 AND 2

X	0	1	2
P(X)	5/18	5/9	1/6

WHEN GREEN BALLS IS DRAWN WE HAVE PROBABILITY DISTRIBUTION AS FOLLOWS. NO. OF GREEN  BALLS DRAWN WILL BE 0,1 AND 2

Y	0	1	2
P(Y)	1/6	5/9	5/18

THE JOINT PROBABILITY IS GIVEN BY

X|Y	0	1	2
0	0	0	5/18
1	0	5/9	0
2	1/6	0	0

We need to calculate the value of Cov(X,Y), E(X²) ,E(Y²),E(X)², E(X)², σ_x and σ_y

(i) E(X) = 0*5/18 + 1*5/9 + 2*1/6

= 5/9 + 1/3

=8/9

E(X)² = 64/81

E(X²) = 0*5/18 + 1*5/9 + 4*1/6

= 5/9 + 2/3

=11/9

E(Y) = 0*1/6 + 1*5/9 + 2* 5/18

= 5/9 + 10/18

= 11/9

E(Y)² = 121/81

E(Y²) = 0*1/6 + 1*5/9 + 4* 5/18

= 5/3

E(X,Y) = 5/9

We need to calculate the value of Cov(X,Y), E(X²) ,E(Y²),E(X)², E(X)², σ_x and σ_y

Cov(X,Y) = E(X,Y) -E(X).E(Y)

= 5/9 -8/9*11/9

= -43/81

σ_x = E(X²) -E(X)²

= 11/9 - 64/81

= 35/81

σ_y = E(Y²) -E(Y)²

= 5/3 - 121/81

= 14/81

CORRELATION COEFFICIENT = Cov(X,Y)/(σ_x.σ_Y)

= (-43/81)/ {(35/81)*(14/81)}

= (-43*81)/(35*14)

= -7.1332