Consider the following situation. Suppose we have two fair dice, D1

with 5 red sides and 1 white side and D2 with 1 red side and 5 white sides.

We pick one of the dice randomly, and throw it repeatedly until red comes

up for the first time. With the same die this experiment is repeated two more

times. Suppose the following happens:

First experiment: first red appears in 3rd throw

Second experiment: first red appears in 5th throw

Third experiment: first red appears in 4th throw.

Show that for die D1 this happens with probability 5.7424 10?8, and for

die D2 the probability with which this happens is 8.9725 10?4. Given these

probabilities, which die do you think we picked.

Let X1, X2, . . . be a sequence of independent and identically distributed

random variables with distributions function F. Define Fn as follows: for any a

Fn(a) = number of Xi in (??, a]

n .

Consider a fixed and introduce the appropriate indicator random variables (as

in Section 13.4). Compute their expectation and variance and show that the

law of large numbers tells us that

limn?? P(|Fn(a) ? F(a)| > ?)=0.

13.8 In Section 13.4 we described how the probability density function

could be recovered from a sequence X1, X2, X3, . . . . We consider the

Gam(2, 1) probability density discussed in the main text and a histogram bar

around the point a = 2. Then f(a) = f(2) = 2e?2 = 0.27 and the estimate

for f(2) is Yn/2h, where Yn as in (13.3).

a. Express the standard deviation of Yn/2h in terms of n and h.

b. Choose h = 0.25. How large should n be (according to Chebyshev's inequality) so that the estimate is within 20% of the "true value", with

probability 80%?

13.9 Let X1, X2, . . . be an independent sequence of U(?1, 1) random

variables and let Tn = 1

n

n

i=1 X2

i . It is claimed that for some a and any

? > 0

limn?? P(|Tn ? a| > ?)=0.

a. Explain how this could be true.

b. Determine a.

13.10 Let Mn be the maximum of n independent U(0, 1) random variables.

a. Derive the exact expression for P(|Mn ? 1| > ?).

Hint: see Section 8.4.

b. Show that limn?? P(|Mn ? 1| > ?) = 0. Can this be derived from Chebyshev's inequality or the law of large numbers?

13.11 For some t > 1, let X be a random variable taking the values 0 and t,

with probabilities

P(X = 0) = 1 ? 1

t and P(X = t) = 1

t

.

Then E[X] = 1 and Var(X) = t?1. Consider the probability P(|X ? 1| > a).

a. Verify the following: if t = 10 and a = 8 then P(|X ? 1| > a)=1/10 and

Chebyshev's inequality gives an upper bound for this probability of 9/64.

The difference is 9/64 ? 1/10 ? 0.04. We will say that for t = 10 the

Chebyshev gap for X at a = 8 is 0.0

Business Statistics Standard Deviation, Grouped Data The net income of a sample of 5&P 500 large stock dividend believes that their net income follow a normal probability distribution. To confirm this, the following were sampled, organized into the following frequency distribution. Based on the distribution, what is the estimate standard deviation? Show all work with Formula. Net Income ($ million] Frequency M EM f ( m - x ) ? Y = $5.50 up to $6.50 20 6.50 up to 7.50 24 7.50 up to 8.50 130 8.50 up to 9.50 68 9.50 up to 10.50 2822. (a) By definition, two events A and B are statistically independent if and only if P(A | B) = P(A). Prove mathematically that two events A and B are independent if and only if P(A | B) = P(A | BC). [Hint: Let P(A) = a, P(B) = b, P(A n B) = c, and use either a Venn diagram or a 2 x 2 table.] (b) More generally, let events A, B1, B2, ..., Bn be defined as in Bayes' Theorem. Prove that: A and B1 are independent, A and B2 are independent, ..., A and B, are independent if and only if P(A | B1) = P(A | B2) = ... = P(A | B,). [Hint: Use the Law of Total Probability.]1. A statistics professor has determined the following probability distribution for X, the grade which a student will earn in a business statistics class. Table for Exercise 1 - Grade Distribution Grade X P(X = x) 4.0 0.15 3.0 0.35 2.0 0.25 1.0 0.15 0.0 0.10 a. What is the average grade that a student will earn in a business statistics class? b. Find the variance of the grades that students will earn in a business statistics class. c. Find the standard deviation of the grades which students will earn in a business statistics class