Question

Two number are selected at random (without replacement but with different arrangement) from the first four positive integers. Let X denote the larger of the two numbers obtained. Find expectation of X.

Answer 1

The sample space is

S = {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)}

∴ n(S) = 12

1 is not larger than any given numbers

2 is larger in two cases i.e. (1, 2), (2, 1)

3 is larger in 4 cases and

4 is larger in 6 cases

∴ The r.v.X can take values 2, 3, 4

P(X = 2) = `2/12 = 1/6`,

P(X = 3) = `4/12 = 1/3`

P(X = 4) = `6/12 = 1/2`

X	2	3	4
P(X)	`1/6`	`1/3`	`1/2`

Here `sump_i = 1/6 + 1/3 + 1/2`

= `(1 + 2 + 3)/6`

= 1

E(X) = `sump_ix_i`

= `2 xx 1/6 + 3 xx 1/3 + 4 xx 1/2`

= `1/3 + 1 + 2`

= `10/3`