∵ two numbers are selected at random like {(2,3) or (5,4) or (4,5)..etc}

Total ways of selecting two numbers without replacement = 6 x 5 =30

As X denote the larger of two numbers selected

∴ X can take values 3,4,5,6 and 7

P(X=3) = P(larger number is 3) = [{2,3},{3,2}]

P(X=4) = P(larger number is 4) = [{2,4},{4,2},{3,4},{4,3}]

P(X=5) = P(larger number is 5) = [{2,5},{3,5},{4,5} and their reverse order]

P(X=6) = P(larger number is 6) = [{2,6},{3,6},{4,6},{5,6} and their reverse order]

P(X=7) = P(larger number is 7) = [{2,7},{3,7},{4,7},{5,7},{6,7} and their reverse order]

Now we have p_i and x_i.

Let’s proceed to find mean and variance.

Mean of any probability distribution is given by Mean = ∑x_ip_i

Variance is given by:

Variance = ∑ x_i²p_i – (∑x_ip_i)²

∴ first we need to find the products i.e. p_ix_i and p_ix_i² and add them to get mean and apply the above formula to get the variance.

Following table representing probability distribution gives the required products :

∴ Mean =

∵ Variance = ∑ x_i²p_i – (∑x_ip_i)²

∴ Variance =