A large urn contains a collection of balls, 1/3 numbered 0, 1/3 numbered 2, and 1/3 numbered 4. An experiment consists of selecting a ball, noting its number, putting the ball back, selecting another, and noting its number. There are 9 distinct ordered pairs of numbers that can be obtained. Namely (0,0), (0,2), (2,2), (2,4), (4,0), (4,2), and (4,4). For example, (2,4) means a 2 was selected first, followed by a 4, Note that x = 3 for the pair (2,4).

(a) Obtain the probability distribution of x.

(b) Obtain U slash/x and O 2/X

Let x = number of the ball obtained on any selection. Then the probability distribution for x is given by:

(c) For this distribution, find (I) u, and (ii) O2 (2^nd power) with formulas from Chapter 5. Compare with part (b), and note that u (slash over x) = u and O 2/n, where n = 2 is the sample size.