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Sampling, Estimation and Simulation.R
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library(fitdistrplus)
library(triangle)
library(psych)
library(nortest)
#--------------------
#Problem 1 :
#--------------------
results = NULL
for (k in 1:5000){
Beth_Israel_Medical <- rtriangle(1, 20, 300, 80)*0.3
Tufts_Medical <- rtriangle(1, 20, 300, 80)*0.15
Massachusetts_General <- rtriangle(1, 20, 300, 80)*0.2
Boston_Medical <- rtriangle(1, 20, 300, 80)*0.25
Brigham_and_Women <- rtriangle(1, 20, 300, 80)*0.1
total = min (Beth_Israel_Medical, Tufts_Medical, Massachusetts_General, Boston_Medical, Brigham_and_Women)
results = rbind(results, data.frame(Beth_Israel_Medical, Tufts_Medical, Massachusetts_General, Boston_Medical, Brigham_and_Women, total))
}
head(results)
summary(results)
result_time <- data.frame(Beth_Israel_Medical_time = results$Beth_Israel_Medical*(7*2/60),
Tufts_Medical_time = results$Tufts_Medical*(10*2/60),
Massachusetts_General_time = results$Massachusetts_General*(15*2/60),
Boston_Medical_time = results$Boston_Medical*(15*2/60),
Brigham_and_Women_time = results$Brigham_and_Women*(20*2/60),
total_time = results$total*(7*2/60))
head(result_time)
head(results)
summary(result_time)
summary(results)
#Histogram
library(ggplot2)
gg_func <- function(s, t = results, k = "Triangular Distribution Histogram"){
ggplot(t, aes(x = t[,c(s)])) +
geom_histogram(colour="black", fill="white", binwidth = 1) +
xlab(toupper(s)) +
ylab("Frequency") +
ggtitle(k)
}
gg_func("Beth_Israel_Medical")
gg_func("Tufts_Medical")
gg_func("Massachusetts_General")
gg_func("Boston_Medical")
gg_func("Brigham_and_Women")
#Law of large number
avg <- 0
n <- 0
for (k in 1:5000) {
x <- mean(sample(results$Beth_Israel_Medical, k, replace = TRUE))
avg[k] = x
n[k] = k
avg_beth <- data.frame(avg, n)
}
#Plot
p <- ggplot(avg_beth, aes(x=n, y=avg)) +
geom_point(shape=1) +
geom_smooth(method=lm, se=FALSE) +
xlab("Sample Size") +
ylab("Sample Mean") +
ggtitle("Beth Israel Medical-Law of Large Numbers")
p
#C.I.
ci <- function(alpha,m,n){
alpha <- (1-(alpha/100))
conf <- 1-alpha
me <- qt(conf, df = n - 1)*(sd(m)/sqrt(n))
cat("Lower Bound", mean(m) - me, '\n')
cat("True Mean", mean(m), '\n')
cat("Uper Bound", mean(m) + me, '\n')
cat("Margin Of Error", me, '\n')
}
ci(95,result_time$Beth_Israel_Medical,5000)
#Histogram
gg_func("Beth_Israel_Medical_time", result_time, "Relative Frequency Histogram")
#Probability Plot
p <- ggplot(result_time, aes(x=Beth_Israel_Medical_time)) +
geom_histogram(aes(y=..density..),
binwidth=1,
colour="black", fill="white") +
geom_density(alpha=.5, fill="#FF6666") +
xlab("Beth_Israel_Medical") +
ylab("Frequency") +
ggtitle("Probability Plot")
p
#Chi-square test
p.tri <- pltriangle(result_time$Beth_Israel_Medical_time)
chisq.test(result_time$Beth_Israel_Medical_time, p = p.tri, rescale.p = T)
#problem 5 with t
t_prob <- function(m,n){
data.frame((m/n * 60)/m)
}
avg_per <- data.frame(t_prob(results$Beth_Israel_Medical, result_time$Beth_Israel_Medical_time),
t_prob(results$Tufts_Medical, result_time$Tufts_Medical_time),
t_prob(results$Massachusetts_General, result_time$Massachusetts_General_time),
t_prob(results$Boston_Medical, result_time$Boston_Medical_time),
t_prob(results$Brigham_and_Women, result_time$Brigham_and_Women_time))
colnames(avg_per) <- c("Beth_Israel_Medical", "Tufts_Medical", "Massachusetts_General", "Boston_Medical", "Brigham_and_Women")
for (i in n) {
avg_per$per_victim[i] <- sum(avg_per$Beth_Israel_Medical[i], avg_per$Tufts_Medical[i], avg_per$Massachusetts_General[i], avg_per$Boston_Medical[i], avg_per$Brigham_and_Women[i])
}
ci(95,avg_per$per_victim,5000)
#Histogram
gg_func("per_victim", avg_per, "Relative Frequency Histogram")
#Probability Plot
p <- ggplot(avg_per, aes(x=per_victim)) +
geom_histogram(aes(y=..density..),
binwidth=1,
colour="black", fill="white") +
geom_density(alpha=.5, fill="#FF6666") +
xlab("Value") +
ylab("Frequency") +
ggtitle("Probability Plot")
p
#Chi-square test
p.tri <- pltriangle(avg_per$per_victim)
chisq.test(avg_per$per_victim, p = p.tri, rescale.p = T)
#--------------------
#Problem 2 :
#--------------------
results = NULL
for (k in 1:5000){
Beth_Israel_Medical <- rnorm(1, 150, 50)*0.3
Tufts_Medical <- rnorm(1, 150, 50)*0.15
Massachusetts_General <- rnorm(1, 150, 50)*0.2
Boston_Medical <- rnorm(1, 150, 50)*0.25
Brigham_and_Women <- rnorm(1, 150, 50)*0.1
total = min (Beth_Israel_Medical, Tufts_Medical, Massachusetts_General, Boston_Medical, Brigham_and_Women)
results = rbind(results, data.frame(Beth_Israel_Medical, Tufts_Medical, Massachusetts_General, Boston_Medical, Brigham_and_Women, total))
}
head(results)
describe(results)
results <- abs(results)
gg_func("Beth_Israel_Medical", k = "Normal Distribution Histogram")
gg_func("Tufts_Medical", k = "Normal Distribution Histogram")
gg_func("Massachusetts_General", k = "Normal Distribution Histogram")
gg_func("Boston_Medical", k = "Normal Distribution Histogram")
gg_func("Brigham_and_Women", k = "Normal Distribution Histogram")
result_time <- data.frame(Beth_Israel_Medical_time = ((results$Beth_Israel_Medical*(7*2/60))+2),
Tufts_Medical_time = ((results$Tufts_Medical*(10*2/60))+4),
Massachusetts_General_time = ((results$Massachusetts_General*(15*2/60))+3),
Boston_Medical_time = ((results$Boston_Medical*(15*2/60))+5),
Brigham_and_Women_time = ((results$Brigham_and_Women*(15*2/60))+5),
total_time = ((results$total*(7*2/60))+2))
#Law of large number
avg <- 0
n <- 0
for (k in 1:5000) {
x <- mean(sample(results$Beth_Israel_Medical, k, replace = TRUE))
avg[k] = x
n[k] = k
avg_beth <- data.frame(avg, n)
}
#Plot
p <- ggplot(avg_beth, aes(x=n, y=avg)) +
geom_point(shape=1) +
geom_smooth(method=lm, se=FALSE) +
xlab("Sample Size") +
ylab("Sample Mean") +
ggtitle("Plot")
p
#C.I.
ci <- function(alpha,m,n){
alpha <- (1-(alpha/100))
conf <- 1-alpha
me <- qt(conf, df = n - 1)*(sd(m)/sqrt(n))
cat("Lower Bound", mean(m) - me, '\n')
cat("True Mean", mean(m), '\n')
cat("Uper Bound", mean(m) + me, '\n')
cat("Margin Of Error", me, '\n')
}
ci(95,result_time$Beth_Israel_Medical_time,5000)
#Histogram
gg_func("Beth_Israel_Medical_time", result_time, "Relative Frequency Histogram")
#Probability Plot
p <- ggplot(result_time, aes(x=Beth_Israel_Medical_time)) +
geom_histogram(aes(y=..density..),
binwidth=1,
colour="black", fill="white") +
geom_density(alpha=.5, fill="#FF6666") +
xlab("Value") +
ylab("Frequency") +
ggtitle("Probability Plot")
p
fit <- fitdistr(result_time$Beth_Israel_Medical_time,"normal")
#Chi-square test
p.norm <- pnorm(result_time$Beth_Israel_Medical_time)
chisq.test(abs(result_time$Beth_Israel_Medical_time), p = p.norm, rescale.p = T)
#problem 5 with t
t_prob <- function(m,n){
data.frame((m/n * 60)/m)
}
avg_per <- data.frame(t_prob(results$Beth_Israel_Medical, result_time$Beth_Israel_Medical_time),
t_prob(results$Tufts_Medical, result_time$Tufts_Medical_time),
t_prob(results$Massachusetts_General, result_time$Massachusetts_General_time),
t_prob(results$Boston_Medical, result_time$Boston_Medical_time),
t_prob(results$Brigham_and_Women, result_time$Brigham_and_Women_time))
colnames(avg_per) <- c("Beth_Israel_Medical", "Tufts_Medical", "Massachusetts_General", "Boston_Medical", "Brigham_and_Women")
for (i in n) {
avg_per$per_victim[i] <- sum(avg_per$Beth_Israel_Medical[i], avg_per$Tufts_Medical[i], avg_per$Massachusetts_General[i], avg_per$Boston_Medical[i], avg_per$Brigham_and_Women[i])
}
ci(95,avg_per$per_victim,5000)
#Histogram
gg_func("per_victim", avg_per, "Relative Frequency Histogram")
#Probability Plot
p <- ggplot(avg_per, aes(x=per_victim)) +
geom_histogram(aes(y=..density..),
binwidth=1,
colour="black", fill="white") +
geom_density(alpha=.5, fill="#FF6666") +
xlab("Value") +
ylab("Frequency") +
ggtitle("Probability Plot")
p
fit <- fitdistr(avg_per$per_victim, 'normal')
#Chi-square test
p.norm <- pnorm(abs(avg_per$per_victim))
chisq.test(avg_per$per_victim, p = p.exp, rescale.p = T)