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set.seed(1)
coin_flips <- rbinom(100, 1, .5)
coin_flips

  [1] 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1
 [51] 0 1 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 1 1 1 0 0 1 1

Data <- data.frame(toss_number = (1:length(coin_flips)) , toss_result = coin_flips)
head(Data, 5)

frequencies <- function(vec) {
  len_vec <- length(vec)
  avg <- 1:len_vec
  
  for (i in 1:len_vec) {
    avg[i] <- length(which(vec[1:i]==1))/i
  }
  return(avg)
}

freq <-frequencies(coin_flips)
Data$avgs <- freq
head(Data, 5)

library(ggplot2)

ggplot(data=Data, aes(x=toss_number, y=avgs, group=1)) +
  geom_line() +
  geom_point()

set.seed(1)

sum <- runif(10000, min = 0, max = 1)

mean(sum)

[1] 0.500168

estimate_pi <-function(seed = 1, iterations = 100000000){
  set.seed(seed)
  x <- runif(n = iterations, min = 0, max = 1)
  val_of_g = sqrt(1-x^2)
  pi_over_four = mean(val_of_g)
  pi <- 4*pi_over_four
  return(pi)
}
estimate_pi()

[1] 3.141566

library('tidyverse')

Registered S3 methods overwritten by 'dbplyr':
  method         from
  print.tbl_lazy     
  print.tbl_sql      
── Attaching packages ───────────────────────────────────────────────────────────────── tidyverse 1.3.2 ──✔ tibble  3.1.8     ✔ dplyr   1.1.0
✔ tidyr   1.3.0     ✔ stringr 1.5.0
✔ readr   2.1.3     ✔ forcats 1.0.0
✔ purrr   1.0.1     ── Conflicts ──────────────────────────────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()

lottery <- function(seed = 1, iterations = 15000) {
  set.seed(seed)
  num1 <- floor(runif(n = iterations, min = 1, max = 4))
  num2 <- floor(runif(n = iterations, min = 1, max = 4))
  num3 <- floor(runif(n = iterations, min = 1, max = 4))
  
  sum <- num1 + num2 + num3
  
  ticket <- 100*num1 + 10*num2 + num3
  ticket <- as.data.frame(ticket)
  
  df <- ticket %>%
    group_by(ticket) %>%
    summarise(counts = n())
  
  df$ticket <- as.factor(df$ticket)
  p <- ggplot(df, aes(x = ticket, y = counts)) +
    geom_bar(fill = "#0073C2FF", stat='identity') +
    theme_bw()
  print(p)
  
  return(list(df, table(sum)))
}

lottery()

[[1]]

[[2]]
sum
   3    4    5    6    7    8    9 
 539 1707 3251 3940 3346 1662  555 

set.seed(40)

#Problem 1
#coin_flips <- rbinom(12, 1, .5)
#coin_flips

a <- (choose(12,7)*(.5^7)*(1-.5)^(12-7))
print(a)

[1] 0.1933594

#a) -> 0.1933594

coin_flips <- rbinom(1000000, 12, .5)
Data <- data.frame(toss_number = (1:length(coin_flips)) , toss_result = coin_flips)
head(Data, 5)

#B

Data_2 <- as.data.frame(table(coin_flips))
head(Data_2, 12)

library(ggplot2)

ggplot(data=Data_2, aes(x=coin_flips, y=Freq, group=1)) +
  geom_line() +
  geom_point()

#C
#This aligns with the frequentist view of probability because we see that over a large number of trials, the frequency of 7 was 193733. 193733/1000000 is .193733. This is close to the number we saw earlier, proving this view.

#D
e_of_x <- 1000000 * .5
ob_avg <- function(vec) {
  len_vec <- length(vec)
  avg <- 0
  
  for (i in 1:len_vec) {
    avg <- vec[i] + avg
  }
  return(avg/len_vec)
}
print(e_of_x/length(coin_flips))

[1] 0.5

ob_avg(coin_flips)

[1] 5.999175

#E
#These observations do match up with the law of large numbers. We can see that these value are very similar, meaning that over a large sample size, the actual mean = expected value, which is the law of large numbers.

#Problem 2

normal_dis <- rnorm(1000000, 4, 1)

Data_n <- data.frame(Number = (1:length(normal_dis)), Result = normal_dis)
head(Data, 10)

#E(x) = 4 by definition of normal distributions
ob_avg(normal_dis)

[1] 4.000485

#These values are very similar, verifying again the law of large numbers for normal distribution

exp_dis <- rexp(1000000, 4)
Data_n <- data.frame(Number = (1:length(exp_dis)), Result = exp_dis)
head(Data, 10)

#E(x) = the integral from 0 to infinity x*lambda*e^-(lambda*x)
#In this case, evaluates to 1/4 or .25
ob_avg(exp_dis)

[1] 0.2503367

#Once again, we have verified the law of large numbers, by acheiving a number very close to the expected value.

#Problem 3-- relative frequency

library('tidyverse')

lottery <- function(seed = 1, iterations = 15000) {
  set.seed(seed)
  num1 <- floor(runif(n = iterations, min = 1, max = 4))
  num2 <- floor(runif(n = iterations, min = 1, max = 4))
  num3 <- floor(runif(n = iterations, min = 1, max = 4))
  
  sum <- num1 + num2 + num3
  
  ticket <- 100*num1 + 10*num2 + num3
  ticket <- as.data.frame(ticket)
  
  df <- ticket %>%
    group_by(ticket) %>%
    summarise(counts = n()/15000)
  
  df$ticket <- as.factor(df$ticket)
  p <- ggplot(df, aes(x = ticket, y = counts)) +
    geom_bar(fill = "#0073C2FF", stat='identity') +
    theme_bw()
  print(p)
  
  return(list(df, table(sum)))
}

lottery()

[[1]]

[[2]]
sum
   3    4    5    6    7    8    9 
 539 1707 3251 3940 3346 1662  555 

set.seed(40)

#problem 4
estimate_pi <-function(seed = 40, iterations = 100000000){
  set.seed(seed)
  x <- runif(n = iterations, min = -1, max = 1)
  val_of_g = sqrt(1-(x^2))
  
  pi_over_two = 2 * mean(val_of_g)
  
  pi <- 2*pi_over_two
  
  return(pi)
}
estimate_pi()

[1] 3.141513

#Problem 5
set.seed(40)

X <- rbinom(10000, 12, .25)
Y <- rbinom(10000, 18, .25)

Z <- rbinom(10000, 30, .25)

Data_x <- as.data.frame(table(X))
head(Data_x, 12)

Data_y <- as.data.frame(table(Y))
head(Data_y, 12)

Data_z <- as.data.frame(table(Z))
head(Data_z, 12)

p <- ggplot(Data_x, aes(x = X, y = Freq/10000)) +
  geom_bar(fill = "#0073C2FF", stat='identity') +
  theme_bw()
print(p)

p2 <- ggplot(Data_y, aes(x = Y, y = Freq/10000)) +
  geom_bar(fill = "#0073C2FF", stat='identity') +
  theme_bw()
print(p2)

p3 <- ggplot(Data_z, aes(x = Z, y = Freq/10000)) +
  geom_bar(fill = "#0073C2FF", stat='identity') +
  theme_bw()
print(p3)

#X has the highest frequency at 3, Y at 4, and Z (ie rbinom(10000, 30, .25)) at 7. 
# By law of large numbers, X and Y will even out to 3 and 4, and so X+Y = 7 = Z. Thus, these frequencies show that they are roughly equivalent.

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