Question

Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X.

Answer 1

∵ 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)

P(X = 4) = P(larger number is 4)

P(X = 5) = P(larger number is 5)

P(X = 6) = P(larger number is 6)

P(X = 7) = P(larger number is 7)

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 :