Question

(1 point) You are to roll a fair die n = 104 times, each time observing the number of dots appearing on the topside of the die. The number of dots showing on the topside of toss i is a random variable represented by Xi, i = 1,2, ..., 104 (a) Consider the distribution of the random variable Xi. Find the mean and the standard deviation of the number of dots showing on the uppermost face of a single roll of this die. (at least one decimal) = (at least three decimals) (b) Now think about the average of your n = 104 tosses of the fair die. What can you say about the distribution of this average? Complete the sentence, using at least three decimals in each numeric answer with a mean The distribution of X ? and a standard deviation o = (c) What is the probability that the average topside on your n = 104 tosses will be somewhere between 3.3 and 3.7? Use at least three decimals in your answer. (d) What is the probability that the average of your n = 104 tosses is greater than 4.05? Enter your answer to at least four decimals.

Answer 1

(a)

For a fair dice, the probability of getting 1, 2, 3, 4, 5 or 6 is 1/6

$\mu_X$ = (1/6) * 1 + (1/6) * 2 + (1/6) * 3 + (1/6) * 4 + (1/6) * 5 + (1/6) * 6 = 3.5

E(X²) = (1/6) * 1² + (1/6) * 2² + (1/6) * 3² + (1/6) * 4² + (1/6) * 5² + (1/6) * 6²

=  15.16667

Var(X) = E(X²) - E(X)²  = 15.16667 - 3.5² = 2.91667

= 1.707826

ox 2.91667

(b)

By Central limit theorem,

= 3.5

L

Standard deviation, = 0.1674661

axo/Vn 1.707826/V104

The distribution of $\bar{X}$ is Normal Distribution with
L
= 3.5 and Standard deviation, $\sigma_{\bar{X}}$ = 0.1674661

(c)

P(3.3 < $\bar{X}$ < 3.7) = P[(3.3 - 3.5)/0.1674661 < Z <  (3.7 - 3.5)/0.1674661]

= P[-1.19 < Z < 1.19]

= P[Z < 1.19] - P[Z < -1.19]

= 0.8830 - 0.1170

= 0.766

(d)

P($\bar{X}$ > 4.05) = P[ Z > (4.05 - 3.5)/0.1674661]

= P[Z >  3.28]

= 0.0005