Question

Q1 (100) Suppose you roll two twenty-five-sided dice. Let X1, X2 the outcomes of the rolls of these two fair dice which can be viewed as a random sample of size 2 from a uniform distribution on integers. a) What is population from which these random samples are drawn? Find the mean (u) and variance of this population (62)? Show your calculations and results.

Answer 1

ANSWER:

a)

Each twenty-sided dice will have the outcome of 1,2,3,.. or 20. Thus, the population is positive integers between 1 and 20.

The probability of any number outcome for a fair dice is 1/20.

Mean,

20 J = E(X) = piti

= (1/20) * 1 + (1/20)* 2 + ... + (1/20) * 20

= (1/20) * [1 + 2 + .. + 20]

= (1/20) * 20 * (20 + 1)/2

Si = n(n+1)/2 17

= 10.5

Now,

Mean,

20 E(X) = Pi*? i=1

= (1/20) * 1² + (1/20)* 2² + ... + (1/20) * 20²

= (1/20) * [1² + 2² + .. + 20²]

= (1/20) * 20 * (20 + 1) * (2 * 20 + 1)/6   

il = n(n+1)(2n + 1)/6

= 143.5

Variance of this population

o? = E(X) - E(X)

= 143.5 - 10.5²

= 33.25