Determine the solutions.

Q.1 Of the voters in Florida, 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.

A hacker is trying to break into a password-protected website by randomly trying to

guess the password. Let m be the number of possible passwords.

(a) Suppose for this part that the hacker makes random guesses (with equal probability),

with replacement. Find the average number of guesses it will take until the hacker guesses

the correct password (including the successful guess).

(b) Now suppose that the hacker guesses randomly, without replacement. Find the average number of guesses it will take until the hacker guesses the correct password (including

the successful guess).

Hint: Use symmetry to find the PMF of the number of guesses.

7. About 60 % of people living in a country wear glasses. When you pick 8 persons randomly from this population, what is the the probability that (a) exactly 2 wear glasses? [Show manual computation.) (b) 7 or more wear glasses. [Show manual computation.] (c) Let X be the number of those who wear glasses among the 8 people selected. What is the range of X? Is the distribution skewed or symmetric? If it is skewed, in what direction? (d) Find the mean and standard deviation of the random variable X. [The answers require very little or no computation.]2. Record the annual depreciation expense for the year 2018. Enter your computation here: 4. Assume that More Company sells the equipment on December 31, 2022 for $350,000. Record the entry to record the sale of the equipment. Use this space to show computation: Use this space to show computation: Use this space to show1. For each type of random variable described in Project 3.1 Step 1, calculate the theoretical mean and variance. Project 3.1 1. Generate a sequence of each of the following types of random variables; each sequence should be at least 10,000 points long- (a) A binomial random variable. Let the number of Bernoulli trials be n = 12. Recall that the binomial random variable is defined as the number of is in in trials for a Bernoulli (binary) random variable. Let the parameter p in the Bernoulli trials be p = 0.5109. (b) A Poisson random variable as a limiting case of the binomial random variable with p = 0.0125 or less and n = 80 or more while maintaining 0 = np = 1. (c) A type 1 geometric random variable with parameter p = 0.09. (d) A (continuous) uniform random variable in the range [-2, 5]. (e) A Gaussian random variable with mean / = 1.3172 and variance o' = 1.9236. (f) An exponential random variable with parameter A = 1.37