Question

Five balls numbered 1,2,3,4, and 5 are placed in an urn. Two balls are randomly selected from the five, and their numbers noted. Let Y be the greater of the two sampled numbers.

a) Find the pdf of Y

b) Find E(Y) and Var(Y)

Answer 1

The total sample space is 5C2 = 10.

Elements of sample space - (1,2) (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5)

a) PDF of Y where Y is the greater of the two sampled numbers.

p(2) = 1/10

p(3) = 2/10

p(4) = 3,10

p(5) = 4/10

b) E(Y) = The expectation of the PDF can be calculated using sum product of the probability and the output values.

So, E(Y) = 1*1/10 + 2*2/10 + 3*3/10 + 4*4/10 = (1 + 4 + 9 + 16)/10 = 30/10 = 3

Var (Y) = E(Y^2) - [E(Y)]^2

E(Y^2) = (1 + 8 + 27 + 64)/10 = 100/10 = 10

Var(Y) = 10 - (3)^2 = 1