Question

Suppose you roll two twenty-sided dice, like the one I gave you
in class today. 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 (µ) and variance of this population (σ2)? Use a Word
File to show your calculations and results.

Answer #1

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,

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

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

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

= 10.5

Now,

Mean,

= (1/20) * 1^{2} + (1/20)* 2^{2} + ... + (1/20)
* 20^{2}

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

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

= 143.5

Variance of this population

= 143.5 - 10.5^{2}

= 33.25

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