coin3 <- data.frame( N_Heads = c(0, 1, 2, 3), C_Heads = c(1, 3, 3, 1) ) attach(coin3) coin3$P_Heads <- C_Heads/sum(C_Heads)

options(repr.plot.width=7, repr.plot.height=4) barplot(coin3$P_Heads, names = coin3$N_Heads, main = "Discrete Probability Distribution of flipping three coins")

# Optional for advanced learners barplot(prop.table(table(c('HHH', 'HHT', 'HHT','HTT','HHT','HTT','HTT', 'TTT'))), main = "Discrete Probability Distribution of tossing three coins", xlab = "x = Number of Heads", ylab = "P(x)", col = "#6633FF")

attach(coin3) E_X = sum(N_Heads * P_Heads) E_X

fees = data.frame( N_Bags = c(0, 1, 2), X = c(0, 25, 60), P_X = c(0.54, 0.34, 0.12) ) fees

attach(fees) E_X <- sum(X * P_X) print(paste("The expected revenue per passenger is $", E_X))

var <- sum((X - E_X)^2 * P_X) print(paste("The variance of the revenue per passenger is ", var))

print(paste("The standard deviation of the revenue per passenger is $", round(sqrt(var),2)))

Uniform_Random_Number <- runif(10000, -2, 1) hist(Uniform_Random_Number, probability = TRUE)

# Optional for Advanced learners a <- -2 b <- 1 rand.unif <- runif(10000, min = -2, max = 1) # Generate 10,000 random numbers between -2 and 1. hist(rand.unif, freq = FALSE, xlab = 'x', ylim = c(0, 0.35), xlim = c(-2.2,1.2), density = 5, main = "Uniform distribution for the interval [-2,1]") curve(dunif(x, min = a, max = b), from = -3, to = 2, n = 100000, col = "darkblue", lwd = 5, add = TRUE, yaxt = "n", ylab = 'probability')

# Demonstration of Normal Distribution plotting options(repr.plot.width=7, repr.plot.height=4) x=seq(-4,4,length=200) y=1/sqrt(2*pi)*exp(-x^2/2) plot(x,y,type="l",lwd=2,col="red", main = "Normal Distribution")

# Optional for advanced learners options(repr.plot.width=7, repr.plot.height=4) n=10; p=0.5; x=0:10; mu=n*p; s=sqrt(n*p*(1-p)) y=dbinom(x,10,p) plot(x,y,type="h",lwd=2,col="red") xx=seq(0,10,length=200) yy=dnorm(xx,mu,s) lines(xx,yy,lwd=2,col="blue")

normal_dist <- function(x, b){ mu = mean(x) sigma = sd(x) x1 <- seq(mu - 6*sigma, mu + 6*sigma, length = 100) # Normal curve fun <- dnorm(x1, mean = mu, sd = sigma) # Histogram options(repr.plot.width=7, repr.plot.height=4) hist(x, prob = TRUE, col = "white", breaks = b, xlim = c(mu - 6*sigma, mu + 6*sigma), ylim = c(0, max(fun)), main = "Histogram overlayed with normal curve") lines(x1, fun, col = 2, lwd = 2) }

data(chickwts) attach(chickwts) normal_dist(chickwts$weight, 10)

shapiro.test(chickwts$weight)

# Is the approximation normal? cat("The data can be approximated as normal distribution since p >", round(shapiro.test(chickwts$weight)$p.value, 2))

chickwts$zscore = (weight - mean(weight))/sd(weight) tail(chickwts)

#b. z_sophiaV = (160 - 151)/7 z_sophiaQ = (157 - 153)/7.67 cat("Z Score for Verbal= ", round(z_sophiaV,2), "\nZ Score for Quantitative Reasoning= ", round(z_sophiaQ, 2))

# e. cat("The verbal ability percentile score is ",round(pnorm(1.29, lower.tail = TRUE),4)*100, "%\n") cat("The quantitative reasoning percentile score is ",round(pnorm(0.52, lower.tail = TRUE),4)*100, "%\n") # f. cat(100 - round(pnorm(1.29, lower.tail = TRUE),4)*100, "%", "did better than Sophia in verbal ability\n") cat(100 - round(pnorm(0.52, lower.tail = TRUE),4)*100, "%", "did better than Sophia in quantitative reasoning")