Find an approximated probability:

Sixteen numbers are selected independently and at random from the interval [ 0, 1 ]. Find an approximated probability that the mean of these sixteen, numbers does not exceed 0.6.

Solution:

Let X_i, i = 1, 2, ..., 16 be the selected numbers. Then X_i are i.i.d. each having a uniform distribution on the interval [O, 1 ]. Hence

μ = E (Xi) =1/2 and σ² = Var Xi =1/12

Hence required probability is

=Φ (1.39) = .9177

The utilization of the central limit theorem in approximating a few probability distributions is given in the subsequent section.