[SOLVED] CS STAT340 HW2: Monte Carlo & Testing

STAT340 HW2: Monte Carlo & Testing

STAT340 HW2: Monte Carlo & Testing

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Instructions

Complete the exercises, update the author and date fields in the header, knit it, and submit both the HTML and RMD files to Canvas. Due date: **Oct 13, 2021 at 11:59pm.

Exercise 1: Generalized birthday problem

[20 points]

The birthday problem asks for the probability that in a group of (n) people, at least 2 people will share the same birthday. This is easy to solve, and the solution is easily found online.

We can generalize this to a more difficult problem and solve it using a Monte Carlo approach: in (n) people, what is the probability that at least (k) people have the same birthday?

Write a function birthday(n,k,i) that returns a probability estimate given 3 arguments:

(n) is the number of people in your sample
for example, if n=50 is used, we are asking in 50 people, what is the probability that

(k) is minimum number of people that must share a birthday
for example if k=4 is used, we asking what is the probability that at least 4 people share the same birthday?

(i) is the number of iterations to run (default 1000)
for example, if i=1000 is used, your function should run 1000 simulations

You may assume there are 365 possible dates (no leap years)
You may assume birthdays are uniformly distributed across the calendar
this is actually not true, see this, or this

You may assume the people are sampled i.i.d

Theres no need to use actual dates in the simulation process. Numbers can represent dates and are easier to generate and manipulate in R. In particular, we recommend using the sample() function with the x, size, and replace arguments set appropriately. See the help page ?sample for details.
Given a vector of numbers, you can easily find duplicates by using the table() function. This will produce a named vector showing how many of each value there are. For example, running table(c(1,3,5,5,7,9,9,9)) will show you there is one 1, one 3, two 5s, one 7, and three 9s.
In your function, you will need to use a for loop to repeat the simulation i times. You will also need a variable outside your for loop to keep track of how many simulations satisfy that # of birthdays (geq k).
If your function is running correctly, then birthday(n=23, k=2), birthday(n=87, k=3) and birthday(n=188, k=4) should all be approximately (50%).
If your function is very slow, consider using the Table function from the Rfast package, which is 4-5 times faster than the normal table() function.

# complete the function
# note i=1000 sets the default value of i to be 1000
birthday = function(n,k,i=1000){
# code goes here

This class currently has 162 enrolled students. What is the approximate probability that at least (4) students have the same birthdays?

ANSWER HERE

Exercise 2: Simulate RV

[15 points]

(X) is a random variable defined between 0 and 1 with the probability density function (f(x)=2x). Note this means the cumulative distribution function is [F(x)=int_0^xf(x)dx=x^2] Write a function rx(n) to sample from this random variable, where n is the size of the sample to be drawn. Then, use your function to draw a sample of 500 and plot a histogram of the output.

# defining pdf of X
pdf_x = Vectorize(function(x){
if(x>0 & x<1){2*x} else 0# showing pdf on plotggplot() + geom_function(fun=pdf_x,n=10001) + theme_minimal() + xlim(c(-1,2)) + ylim(-1,3) + labs(x=’x’,y=’f(x)’)# complete the functionrx = function(n){# code goes here# uncomment the following line of code and check it looks correct# hist(rx(500))Exercise 3: Testing coin flips[15 points]In the six sequences below, only one of them is actually randomly generated from a fair coin. Use a combination of everything you know (common sense, monte carlo, hypothesis testing, etc.) to identify which is actually random and explain your reasoning.flips1 = “HTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHHTHTHTHTHTHTHTTHTHTHTHTHTHTHHTHTHTHTHTHTHTHTHTHTHTHTHTHTHHTTHTHTHTHTHTHTHTHTHTHTHTHTHHTHTHTHTHTHTHTHTHTHTHTHTTHTHTHTHTHTHTHTHTHTHTHTHTHHTHTHTHTHTHTHTHTHTHTHTHHTHTHTHTH”flips2 = “HHHTHTTTHHTHHTHHHTTTTHTHTHHTTHTHHHTHHTHTTTHTHHHTHTTTHTHTHHTHTHTTHTHHTHTHTTTHTHHHTHTHTTHTHTHHTHTHTHHHTHTTTHTHHTHTHTHHTTTHTHHTHHTTTTHTHTHHHTHTTHTHHTHTHTTHTHHTHTHHHTHHHTHTTTHTTHTTTHTHHHTHTHTTHTHHTHHTHTTT”flips3 = “HHTHTHTTTHTHHHTHHTTTHTHHTHTTTHTHTHHTHTHTTHTHHHHHHTTTHTHTHHTHTTTHTHHTHTHTTTHTHHHTTHTTTHTHTHHHHTHTTHHTTTTTHTHHHTHTHTTTTTHHHTHHTHHTHHHTTTTHTHTHHHTHHTTTTTHTHHHTHTHTHTTTHTHHHTHTHTHTTHTHHTHTHTHTTTTHTHHHTHTH”flips4 = “HTHHHHHHHTHTTHHTTHHHTHTHTTTHHTHHHTHHTTHTTTTTTTTTHTHHTTTTTHTHTHTHHTTHTTHTTTTTHHHTHTTTHTHTHHHTHTTTTHTHTHHTTHTHTTHHTHTHHHHTHTTHHTTHTTHTTHTHHHHHHTTTTTTHHHTTHTHHHHTTTHTTHHHTTHTHHTTTHHTHHTTTHTHHTHHHTHHTTHHH”flips5 = “HHHHHHHHHHHTTTTTTTTTTTHHHHHHHHHHHHTTTTTTTTTTTHHHHHHHHHHHHHTTTTTTTTTTHHHHHHHHHHTTTTTTTTHHHHHHHHTTTTTTTHHHHHHHHHTTTTTTTTTHHHHHHHHTTTHHHHHHHHHHHTTTTTTTTTTTHHHHHHHHHHHHTTTTTTTTTTTHHHHHHHHHHHHHTTTTTTTTTTHH”flips6 = “TTHTTTHTTTTTTTHTHTHTHTTHTTHTHHTHHTTTHHTHTTTHTHHTHHHTHTTHHTHHTTHTHTTTTHTHTTTHHTTTTTTTTHTHHTTHTTTTTTHTHTHTHTTTHTTHHTTHTTTHHTTTHTTHTTTTHTTTTHHTTTHTHTHHHTTTTTTHTHHTTTTTTTTTTTTHHHTTTHHHTTTHTTTHTHTTHTTTTTHT”# you can use the function below to split the above sequences in vectors of flipssplit = function(str) strsplit(str, split=””)[[1]]split(flips1) # [1] “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” #[24] “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” #[47] “T” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “H” “T” “H” “T” “H” “T” “H” “T” #[70] “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “H” “T” #[93] “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” # [116] “H” “T” “H” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” # [139] “H” “T” “H” “T” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” # [162] “H” “T” “H” “T” “H” “T” “H” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” “H” “T” # [185] “H” “T” “H” “T” “H” “T” “H” “H” “T” “H” “T” “H” “T” “H” “T” “H”Response goes here: CS: assignmentchef QQ: 1823890830 Email: [email protected]