Of the voters in Canada, a proportion p will vote for candidate G,

and a proportion 1 ? p will vote for candidate B. In an election poll a number

of voters are asked for whom they will vote. Let Xi be the indicator random

variable for the event "the ith person interviewed will vote for G." A model

for the election poll is that the people to be interviewed are selected in such

a way that the indicator random variables X1, X2,. . . are independent and

have a Ber (p) distribution.

a. Suppose we use Xn to predict p. According to Chebyshev's inequality, how

large should n be (how many people should be interviewed) such that the

probability that Xn is within 0.2 of the "true" p is at least 0.9?

Hint: solve this first for p = 1/2, and use that p(1 ? p) ? 1/4 for all

0 ? p ? 1.

b. Answer the same question, but now Xn should be within 0.1 of p.

c. Answer the question from part a, but now the probability should be at

least 0.95.

d. If p > 1/2 candidate G wins; if Xn > 1/2 you predict that G will win.

Find an n (as small as you can) such that the probability that you predict

correctly is at least 0.9, if in fact p = 0.6.

13.5 You are trying to determine the melting point of a new material, of

which you have a large number of samples. For each sample that you measure

you find a value close to the actual melting point c but corrupted with a

measurement error. We model this with random variables:

Mi = c + Ui

where Mi is the measured value in degree Kelvin, and Ui is the occurring

random error. It is known that E[Ui] = 0 and Var(Ui) = 3, for each i, and that

we may consider the random variables M1, M2, . . . independent. According

to Chebyshev's inequality, how many samples do you need to measure to be

90% sure that the average of the measurements is within half a degree of c?

13.6 The casino La bella Fortuna is for sale and you think you might want

to buy it, but you want to know how much money you are going to make. All

the present owner can tell you is that the roulette game Red or Black is played

about 1000 times a night, 365 days a year. Each time it is played you have

probability 19/37 of winning the player's bet of

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.

Consider the triangle kernel, K(u), plotted below. 1.00 - 0.75 - K(u) 0.50 0.25 - 0.00 - -2 -1 N CO The triangle kernel is to be used as the basis for a kernel density estimate. As a first step in its construction, for each x in the data we construct , K (U Zi ). Suppose n = 3 and the data are x1 = 2, 12 = 4, x3 = 7 and use h = 0.3. Illustrate the result of this first step on the axes below.mean; median; mid-range; mode; quart ple variance; standard deviation; coefficient of outlier; inter-quartile range; symmetry; skewn tosis; normal distribution; box plot; histogra and leaf display; kernel density estimate; line chart; column chart; pie chart. 2.6 Exercises 1. Obtain Ezz for the following cases: (a) zi = 1 and n = 5. (b) zi = i and n = 5. (c) zi = 272 and n = 5. (d) zi = 1/i and n = 5.3. Suppose X1,X2,.. .,Xn are iid Generalized Error Distribution GED(D,1,1.5) and you use a kernel density estimate with a rectangular kernel. (:21) Analytically derive an expression for the expected value of the kernel density estimate fb(m) in terms of the odf of GED(0, 1, 1.5). Hint: You can use that the width, to, of rectangular kernel with bandwidth b is approximately w = b - 3.464 , and also note that rectangular kernel with bandwidth parameter b and width to satises that i e s m s s otherwise (b) Based on (a), derive an expression for the bias in terms of edf and pdf of the appropriate Generalized Error Distribution, and then plot the bias of the kernel density estimate as a function of a: for b = .2, .4, .6