¶ Homework 3 ¶

Question 1. Consider the Binomial distribution with \(n = 24\) and \(p = .9\) that is used to model the number of correctly received bits on a satellite link that transmits data in \(24\)-bit blocks.

Quesion 1 (a) Plot the probability density function and cumulative distribution function for the number of correctly received bits

#Answer 1a
n<- 24
p<- 0.9
x<- 0:n
pdf<- dbinom(x, size = n, prob = p)
cdf<- pbinom(x, size = n, prob = p)

#plot probability density function
plot(x, pdf, type = "h", ylim = c(0, max(pdf)))

#plot cumulative distubution function
plot(x, cdf, type = "s", main = "Binomial Distribution CDF")

Quesion 1 (b) What is the mean number of correctly received bits? What is the standard deviation?

#Answer 1b
#(make sure that what you print make sense to me) - for example to print mean you can use cat command once you done with the calculation and have value in the variable
mean_value= n * p
std_dev= sqrt(n * p * (1-p))
cat("Mean number of correctly received bits is ",  mean_value)

Mean number of correctly received bits is  21.6

cat("\n")

cat("The standard deviation is ",  std_dev)

The standard deviation is  1.469694

Quesion 1 (c) What is the probability of more than \(3\)-bit errors in the block of \(24\)?

#Answer 1c
moreThanThree <- 1 - pbinom(3,size=n, prob=p)
moreThanThree

[1] 1

Quesion 1 (d) What is the median of this distribution? What can you interpret from the median value?

#Answer 1d
medianVal<- qbinom(0.5, size = n, prob = p, lower.tail = TRUE)
medianVal

[1] 22

cat("The Median tells us that 50% of the time, we can expect to get 22 or fewer bits correctly. ")

The Median tells us that 50% of the time, we can expect to get 22 or fewer bits correctly. 

Quesion 1 (e) What is the 60th quantile of this distribution? What can you interpret from this value?

#Answer 1e
quantile(n, probs = c(.60))

60% 
 24 

#Interpretation
cat("This tells us that 60% of the time, we can get 24 or fewer bits correctly")

This tells us that 60% of the time, we can get 24 or fewer bits correctly

Question2. The Public Service Answering Point (PSAP) in San Francisco employs \(19\) operators in \(8\)-hour shifts to process \(911\) calls. There are at least \(5\) operators always answering calls. The number of calls processed per operator can be modeled with a Poisson random variable with rate \(\lambda_0 =20\) calls per hour.

Quesion 2 (a) What is the probability an operator can process 20 calls in an hour? Repeat for 30 calls in an hour?

#Answer 2a
lambda<- 20
probTwenty<- dpois(20,lambda)
probThirty<- dpois(30, lambda)

probTwenty

[1] 0.08883532

probThirty

[1] 0.008343536

Quesion 2 (b) Given that 240 calls occurred in an hour, what is the probability that the ten operators can process them all assuming they are split equally among the operators?

#Answer 2b
prob24Calls<- dpois(24, lambda)
prob240Calls<- prob24Calls^10
prob240Calls

[1] 2.892317e-13

Quesion 2 (c) Now, If the aggregate call rate per hour \(\lambda_c\) , is measured by \(\lambda_c = 85\) calls per hour, what is the probability of more than \(100\) calls in an hour

#Answer 2c
lambda_c<- 85
moreThan100<- 1-ppois(100,lambda_c)
moreThan100

[1] 0.04934533

Question 3. For the two random number generator below A and B(don’t forget to add your R code)

- [A] \(Z_i = (9Z_{i-1} + 1) \mod 16\) with \(Z_0 = 5\).

- [B] \(Z_i = (7Z_{i-1} + 3) \mod 32\) with \(Z_0 = 10\),

Quesion 3 (a) Compute \(Z_i\) and \(U_i\) for values of \(i\) until a number is repeated, what is the period of both the generator? Provide your comments about the period of both RGN

#Answer 3a
Z_A <- 5
periodA <- 0
U_A_values <- numeric()

Z_B <- 10
periodB <- 0
U_B_values <- numeric()

scatter_A <- list()
scatter_B <- list()

nextA <- function(Z) {
  return((9 * Z + 1) %% 16)
}
nextB <- function(Z) {
  return((7 * Z + 3) %% 32)
}

while (TRUE) {
  Z_nextA <- nextA(Z_A)
  U_A <- Z_nextA / 16
  periodA <- periodA + 1
  U_A_values <- c(U_A_values, U_A)
  scatter_A[[periodA]] <- U_A_values
  
  if (Z_nextA %in% U_A_values) {
    break
  }
  
  Z_A <- Z_nextA
  
  Z_nextB <- nextB(Z_B)
  U_B <- Z_nextB / 32
  periodB <- periodB + 1
  U_B_values <- c(U_B_values, U_B)
  scatter_B[[periodB]] <- U_B_values
  
  if (Z_nextB %in% U_B_values) {
    break
  }
  
  Z_B <- Z_nextB
  


}
cat("Period A: ", periodA)

Period A:  11

cat("\n")

cat("Period B: ", periodB)

Period B:  10

The period is the number of values that were generated before a repeat occured.

Quesion 3 (b) Which of these parameters effect the period of LCG – \(a\), \(b\), \(Z_0\)

#Answer 
cat("The parameters a and b can affect the period. Changing these values can vary the amount of values that are generated before a repeat.")

Quesion 3 (c) For both generators plot a scatter diagram of the Zi values 1 apart. What are your observations from these plots? What do you think about the property of randomness of these generators?

#Answer
plot(unlist(scatter_A[[1]]), unlist(scatter_A[[2]]), main = "Scatter Diagram A")

Error in xy.coords(x, y, xlabel, ylabel, log) : 
  'x' and 'y' lengths differ

Quesion 3 (d) Randomness of default random generator of R – Run runif command to generate 100 random numbers and plot the scatter diagram of these numbers ( values 1 apart) and discuss your observations about randomness of this random generator.

#Answer
random_numbers <- runif(100)
plot(random_numbers[-100], random_numbers[-1], main = "Scatter Diagram R")

cat("The randomness of this generator is truly random as seen in the scatter plot. The values are dispersed across the whole range of Scatter Diagram R")

The randomness of this generator is truly random as seen in the scatter plot. The values are dispersed across the whole range of Scatter Diagram R

Quesion 3 (e) Compute the mean value of \(U_i\) across the period

#Answer
meanA <- mean(U_A_values)
meanB <- mean(U_B_values)
meanA

[1] 0.5056818

meanB

[1] 0.371875

Quesion 3 (f) By providing a plot of density (histogram) discuss the uniformity of both of the generators

#Answer
hist(U_A_values, main = "Histogram of Generator A", xlim = c(0, 1))

hist(U_B_values, main = "Histogram of Generator B", xlim = c(0, 1))

cat("While the frequencies between the two generators changes for some values, the frequencies of both are still within the same range. ")

While the frequencies between the two generators changes for some values, the frequencies of both are still within the same range. 

Question 4. Using the inverse transform method

Question 4.(a) Develop an algorithm for the random variable with cumulative distribution function F(x) below.

\[ F(x) = 1 - e^{-(x/\lambda)^k} \]

where \(x \geq 0\), \(\lambda \geq 0\), and \(k \geq 0\).

# Answer 4a
# This might be hard to print using R - you can write the step in your notebook and then include image here / or you can include as seperate file when upload

# Check if the IRdisplay package is already installed
if (!require(IRdisplay, quietly = TRUE)) {
  # If not installed, install it
  install.packages("IRdisplay")
  
  # Load the IRdisplay library
  library(IRdisplay)
} else {
  # If already installed, just load the library
  library(IRdisplay)
}


# Embed an image
#display_png(file = "example.png")

Quesion 4 (b) Use the first three Ui values from part (a) and generator [A] of previous problem to create 3 values from the random variable when, \(\lambda = 1\), \(𝑘 = 5\)

#Answer

Quesion 4 (c) sing R generate 10,000 values from your algorithm when = 1 , 𝑘 = 5 and plot a histogram of the density. Discuss what insights you obtain when you look at the plot generated here vs the plot you generated in question 3 (e)

#Answer

Question 5. Develop a Monte Carlo simulation in R that counts the number of uniform [0,1] random numbers that must be summed to get a sum greater than 1. Run a single simulation with n = 10,000 times and find the mean of the number of counts. Do you think this number looks somewhat familiar. Every student might get slightly different value any ideas why?

#Answer
sims<- 1000
counts<- numeric(sims)

for(i in 1:sims){
  sum<-0
  count<-0
  
  while(sum<=1){
    rand<-runif(1)
    sum<-sum+rand
    count<- count+1
  }
  counts[i]<-count
  
  meanOfCount<-mean(counts)
  cat("mean number of counts: ", meanOfCount, "\n")
 
}

mean number of counts:  0.002 
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mean number of counts:  1.361 
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mean number of counts:  1.365 
mean number of counts:  1.368 
mean number of counts:  1.371 
mean number of counts:  1.373 
mean number of counts:  1.376 
mean number of counts:  1.38 
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mean number of counts:  2.61 
mean number of counts:  2.612 
mean number of counts:  2.616 
mean number of counts:  2.619 
mean number of counts:  2.622 
mean number of counts:  2.625 
mean number of counts:  2.628 
mean number of counts:  2.63 
mean number of counts:  2.632 
mean number of counts:  2.634 
mean number of counts:  2.636 
mean number of counts:  2.638 
mean number of counts:  2.64 
mean number of counts:  2.642 
mean number of counts:  2.644 
mean number of counts:  2.646 
mean number of counts:  2.651 
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mean number of counts:  2.656 
mean number of counts:  2.659 
mean number of counts:  2.661 
mean number of counts:  2.667 
mean number of counts:  2.671 
mean number of counts:  2.674 
mean number of counts:  2.677 
mean number of counts:  2.68 
mean number of counts:  2.682 
mean number of counts:  2.684 
mean number of counts:  2.686 
mean number of counts:  2.689 
mean number of counts:  2.693 
mean number of counts:  2.696 
mean number of counts:  2.7 
mean number of counts:  2.703 
mean number of counts:  2.705 
mean number of counts:  2.707 
mean number of counts:  2.709 
mean number of counts:  2.712 
mean number of counts:  2.715 
mean number of counts:  2.718 
mean number of counts:  2.723 
mean number of counts:  2.725 
mean number of counts:  2.727 
mean number of counts:  2.731 
mean number of counts:  2.734 
mean number of counts:  2.736 
mean number of counts:  2.738 
mean number of counts:  2.741 
mean number of counts:  2.743 
mean number of counts:  2.746 
mean number of counts:  2.749 
mean number of counts:  2.751 
mean number of counts:  2.754 

cat("\n")

cat("Different students may get different values because the Monte Carlo simulation uses random numbers. So every individual may get different random values thus making the values vary from computer to computer. ")

Different students may get different values because the Monte Carlo simulation uses random numbers. So every individual may get different random values thus making the values vary from computer to computer. 

Question 6. For our specific random variable, we know that its PDF is defined by the function \(f(x) = 10x(1-x)\). Utilize the accept-reject algorithm to draw samples from this distribution. Take a look at Lecture 10 slide 8 and 9

#Answer
fx<- function(x) {
  return(10 * x * (1 - x))
}
M<- 2.5
generate_sample <- function() {
  while(TRUE) {
    x<- runif(1)
    y <- runif(1, max = M)

    if (y <= fx(x)) {
      return(x) 
    }
  }
}

set.seed(12345) 
num_samples<- 1000
samples<- replicate(num_samples, generate_sample())

hist(samples, breaks = 30, probability = TRUE,
     main = "Histogram of Samples from f(x) = 10x(1-x)")


curve(fx(x), col = "red", lwd = 2, add = TRUE)

legend("topright", legend = c("Generated Samples", "f(x)"), col = c("blue", "red"), lwd = 2)