Question

Two the numbers are selected at random (without replacement) from first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution.

Answer 1

First six positive integers are {1, 2, 3, 4, 5, 6}
No. of ways of selecting 2 numbers from 6 numbers without replacement = ⁶C₂ = 15
X denotes the larger of the two numbers, so X can take the values 2, 3, 4, 5, 6.
Probability distribution of X:

X	2	3	4	5	6
P(x)	1/15	2/15	3/15	4/15	5/15

Computation of Mean and Variance:

Xi	P(X=xi)	pixi	pixi2
2	1/15	2/15	4/15
3	2/15	6/15	18/15
4	3/15	12/15	48/15
5	4/15	20/15	100/15
6	5/15	30/15	180/15
		`sum p_ix_1=70/15=14/3`	`sump_ix_^2=350/15=70/3`

 

Mean `=sump_ix_i=70/15=4.67`

Variance `=sump_ix_i^2-(sump_ix_i)^2=70/3-196/9=(210-196)/9=14/9`