Question

Consider a fair six-sided die. (a) What is its probability mass function? Graph it. It represents the population distribution of outcomes of rolls of a six-sided die (b) How would you describe the population distribution? (c) What is the sampling distribution of x for a six-sided fair die, when its rolled 100 times? Describe it with as much specificity as possible. NOTE: Roll of a die is a discrete variable. Why is it ok to use the Normal distribution to describe the sampling distributiorn of the average roll for a die?

Answer 1

a)

let X be the random variable denoting a number of fair die thrown. 1 4 5 6 Total P(Xx1/61/61/61/61/61/6 1

b)

Scatterplot of probability vs x 0.22 0.20 0.18 O 0.16 0.14 0.12 4

the distribution is uniform to the all points

c) here we can see the expectation and variance of X is

E(X)=3.5

var(X)=2.91666666666667

"X 2 Vaa (x) E(45 ニ2.91667 Sat 2.91

so the variance and mean is constant hence we can apply the central limit theorem for the large number of samples we have observed.

Distribution Plot Normal, Mean-3.5, StDev-0.170783 2.0 1.5 1.0 0.5 0.0 3.0 3.2 3.4 3.6 3.8 4.0