Discrete Random Variable

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")

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")

Expected Value

attach(coin3)

E_X = sum(N_Heads * P_Heads) cat("Expected value of number of heads in tossing 3 coins is ", E_X)

Variance

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) cat("The expected revenue per passenger is $", E_X)

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

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

Binomial Distribution

x <- seq(0,50) y <- dbinom(x,50,0.5) plot(x, y, 'h')

Continuous Random Variable

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

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')

Normal Distribution

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")

Normal Approximation

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, 15)

Test of Normality

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))

Z Score

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

Problems from the book

Solution

a NVerbal:(μ=151,σ=7), NQuantitativeReasoning:(μ=153,σ=7.67)

#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))

c. She scored 1.29 standard deviations above the mean on the Verbal Reasoning section and 0.52 standard deviations above the mean on the Quantitative Reasoning section. d. She did better on the Verbal Reasoning section since her Z score on that section was higher.

# 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")

g. We cannot compare the raw scores since they are on different scales. Her scores will be measured relative to the merits of other students on each exam, so it is helpful to consider the Z score. Comparing her percentile scores is a more appropriate way of determining how well she did compared to others taking the exams.h. Answer to part (b) would not change as Z scores can be calculated for distributions that are not normal. However, we could not answer parts (c)-(f) since we cannot use the Z table to calculate probabilities and percentiles without a normal model.

.css-15w88e5{color:var(--chakra-colors-fg-neutral-primary);font-weight:inherit;letter-spacing:-0.09px;}Discrete Random Variable