ANLY-511代写、代做R程序语言、代写R、代做symmetric random

日期：2019-09-30 10:12

ANLY-511 Homework 4 Problems

Submit problems 35,37,38,40,41,42 and 46. Explain your work, give concise

reasoning, and . Attach R code with comments if applicable. Using Markdown

is the best way to do this. Do not print out any data or any detailed results of

simulations.

35. (2 points) Consider a symmetric random walk: the states are integers

x ∈ Z and the transition probabilities are

P(Xi+1 = x + 1|Xi = x) = P(Xi+1 = x − 1|Xi = x) = 1

2

for all i and all x. Assume that X0 = 0. Let T be the random time when |XT | = 15

for the first time. Use a simulation to generate a few hundred values of T and then

make a box plot of T. Your answer should consist of commented simulation code

and either the box plot or a description (max, min, quartiles, median). Also state

the observed probability of not hitting x = ±15 at all.

36. (2 points) In American roulette it is possible to bet on a block of twelve

numbers, consisting e.g. of the numbers from 1 to 12 or from 13 to 24. The casino

will pay you 2:1, i.e. three times your bet if one of these numbers comes up, and

you lose your bet otherwise. Propose a modification of the St. Petersburg system

for somebody who only uses this bet, and explain it. The first bet and the bet after

each win should be $1. If you win after several losses, you should win back all your

losses plus some extra money. Assume that you can make bets in dollars and cents.

37. (2 points) Bob’s preferred bet in American roulette consists in betting

$1 on black numbers and simultaneously $2 on even numbers (see the roulette

board in the course slides). Find all possible outcomes of a single game and their

probabilities, that is, find the probability distribution of the outcome of a single

bet. Then compute its expected value.

38. (2 points) Consider a room that is paved with n × n square tiles which

are labeled from 1 to n

2

. The tiles are labeled from 1 to n

2

in some order. A frog

performs a random walk by hopping from one tile to a randomly chosen adjacent

tile in each time step. All adjacent tiles are chosen with the same probability. The

frog can never hop into a wall of the room.

True or not true: the transition matrix for this random walk is symmetric, that is,

it satisfies P(Xi+1 = k|Xi = j) = P(Xi+1 = j|Xi = k) for all i and all possible

states 1 ≤ j, k ≤ n

2

. Explain your answer.

39. (2 points) Suppose X = (X1, X2, X3) has a multinomial distribution with

size n = 10 and probabilities p1 = .2, p2 = .5, p3 = .3. Show that P(X1 = 2, X2 =

4, X3 = 4) = 0.0637875, using the formula involving n! in the slides. Do not use

dmultinom.

40. (2 points) Suppose X = (X1, X2, X3) has a multinomial distribution with

size n = 10 and probabilities p1 = .2, p2 = .5, p3 = .3. Use a simulation with

rmultinom to show that P(X1 = 2, X2 = 4, X3 = 4) ≈ 0.0638. Confirm your

results using dmultinom.

41. (2 points) Suppose X = (X1, X2, X3) has a multinomial distribution with

size n = 10 and probabilities p1 = .2, p2 = .5, p3 = .3. Use a simulation with sample

1

2

(not rmultinom) to show that P(X1 = 2, X2 = 4, X3 = 4) ≈ 0.0638. Confirm your

results using dmultinom.

42. (2 points) Updated on 10/5 Consider the following game: you get to

roll an n-sided fair die once. Let k be the outcome of that die roll, then you get to

throw k darts at a target. The probability of hitting the target is p. Dart throws

are independent of one another.

Let X be the result of the roll of the die, and let Y be the number of hits when you

throw the darts. Set up the joint probability mass function (pmf) for X and Y .

43. (5 points) Consider a symmetric random work as in problem 35. Given

integers a < 0 < b, we are interested in the event that the random walk reaches

x = a before it reaches x = b. Formerly, this means

min{i : Xi = a} < min{j : Xj = b} .

a) Write a function myHits(N,a,b) that simulates N steps of this random walk.

The function then should return T if the simulation reaches x = a before it reaches

x = b (including the case when it reaches x = a but not x = b). It should return F

if the simulated walk reaches x = b before it reaches x = a. It should return NA if

the simulated walk does not reach x = a or x = b during these N steps. Explain

the code.

b) Use a function and sufficiently many simulations to estimate the probability that

a symmetric random walk reaches a = −8 before it reaches b = 6.

44. (5 points) Suppose X = (X1, X2, . . . , X8) has a multinomial distribution

with size n = 10 and probabilities p1 = p2 = p3 = p4 = .2, p5 = p6 = p7 = p8 =

0.05. Use suitable simulations to estimate the following probabilities:

a) P(X1 + X2 + X3 ≤ 5)

b) P(X1 + X2 + X3 ≤ 5 and X6 + X7 + X8 ≥ 3)

c) P(X1 + X2 + X3 ≤ 5 | X1 + X2 + X3 + X4 is even)

45. (5 points) Suppose (X1, X2, X3) is a vector of random variables with the

joint probability density function

f(x1, x2, x3) = (

1 if 0 ≤ x1, x2, x3 ≤ 1

0 otherwise

This is a uniform distribution on the set {(x1, x2, x3) : 0 ≤ x1, x2, x3 ≤ 1}. You

can draw a single sample with the command runif(3).

a) Explain geometrically why P(X2

1 + X2

2 + X2

3 ≤ 1) = π

6

. Think about the volume

of a sphere.

b) Confirm this with a simulation with at least 105

samples.

46. (5 points) Consider the random walk performed by the caveman in the

class slides.

a) Using the transition matrix that was derived in class, compute P(X3 = 3|X0 =

1). Then do the same computation directly. Do not print out the entire transition

matrix.

b) Find the first time T such that the chance of the caveman’s survival for more

than T steps is less than 25% no matter where he starts, using R .

3

47. (5 points) Let N be a random variable with a Poisson distribution with

parameter λ > 0. Given that N = n, let X be a binomial B(n, p) distribution

where 0 < p < 1.

a) Set up the joint probability mass function for N and X, in terms of the parameters

λ and p.

b) Write an R function with input λ, p, k that simulates k values of X.

c) Pick some values of λ and p and simulate sufficiently many instances in each

case to obtain an estimate of E(X). Use sapply or replicate, do not use for

loops. Then guess a formula for E(X) and explain why it makes sense to you. To

document this, only state your choice of λ and p, the number of simulations, and

your estimate for the expected value in each case.