Question

From a sack of fruit containing

3

bananasbananas​,

2

orangesoranges​,

and

2

applesapples​,

a random sample of

4

pieces of fruit is selected. Suppose X is the number of

bananasbananas

and Y is the number of

orangesoranges

in the sample.

​(a) Find the joint probability distribution of X and Y.

​(b) Find

​P[(X,Y)is an element of∈​A],

where A is the region that is given by

StartSet left parenthesis x comma y right parenthesis | x plus y less than or equals 2 EndSet{(x,y) | x+y≤2}.

​(a) Complete the joint probability distribution below.

​(Type integers or simplified​ fractions.)

		x
	​f(x,y)	0	1	2	3	4
yy	0	nothing	nothing	nothing	nothing	nothing

1

nothing

nothing

nothing

nothing

nothing

2

nothing

nothing

nothing

nothing

nothing

3

nothing

nothing

nothing

nothing

nothing

4

nothing

nothing

nothing

nothing

nothing

​(b) The probability is?

Answer 1

X can take values as 0,1,2,3 and Y can take values as 0,1,2

Total number of fruits is 3+2+2=7

Number of ways in which 4 fruits can be choosen from these 7 is

Now the joint PDF is

Y       X->	0	1	2	3
0	0	0
1
2				0

b) Since

Hence