4 The Normal Distributions
4.1 The Probability Density Function
d* series of functions.
dnorm(x = x, mu = mu, sigma = sigma) calculates \(P(X = x)\), where \(X \sim Norm(\mu, \sigma)\).
Probability densities are mostly used to draw the plot of the normal distribution. Instead, we use the CDF for pretty much everything.
4.2 The Cumulative Distribution Function
p* series of functions.
pnorm(x = x, mu = mu, sigma = sigma) calculates \(P(X \le x)\), where \(X \sim Norm(\mu, \sigma)\).
Solution for Example 3.4.1:
Standard Normal Probabilities
I have built a friendly app to help calculate standard normal probabilities. Feel free to bookmark this standalone version and use it for assignment questions (but be aware you’ll need to use the Z-table for the midterm/exam)!
#| standalone: true
#| viewerHeight: 700
# pnorm
library(shiny)
library(ggplot2)
ui <- fluidPage(
# Application title
titlePanel("Normal Probabilities"),
sidebarLayout(
sidebarPanel(
sliderInput("lohi",
"Z values (-4 for -Inf)",
min = -4,
max = 4,
value = c(-4, 1.96),
step = 0.01)
),
mainPanel(
plotOutput("distPlot")
)
)
)
server <- function(input, output) {
output$distPlot <- renderPlot({
lo <- input$lohi[1]
hi <- input$lohi[2]
xseq <- seq(-4,4,0.01)
yseq <- dnorm(xseq)
xrib <- seq(lo, hi, 0.01)
yrib <- dnorm(xrib)
if(lo > -4 & hi < 4){
mytitle <- paste0("pnorm(", hi, ") - pnorm(",lo,") = ",
round(pnorm(hi) - pnorm(lo), 4))
addtox <- c(lo, hi)
arrows <- list(
annotate("segment", x = c(lo, hi), y = c(-0.05, -0.1), size = 0.8,
colour = "darkorchid",
xend = c(-4,-4), yend = c(-0.05, -0.1), arrow = arrow()),
annotate("text", x = mean(c(lo, -4)), y = -0.05, vjust = -0.5,
label = paste0("pnorm(", lo, ")=", round(pnorm(lo), 3))),
annotate("text", x = mean(c(hi, -4)), y = -0.1, vjust = -0.5,
label = paste0("pnorm(", hi, ")=", round(pnorm(hi), 3)))
)
} else if(hi < 4){
mytitle <- paste0("pnorm(", hi, ") = ", round(pnorm(hi), 4))
addtox <- hi
ydouble <- c(0,0)
arrows <- list(
annotate("segment", x = hi, y = -0.05, size = 0.8,
colour = "darkorchid",
xend = -4, yend = -0.05, arrow = arrow()),
annotate("text", x = mean(c(hi, -4)), y = -0.05, vjust = -0.5,
label = paste0("pnorm(", hi, ")=", round(pnorm(hi), 3)))
)
} else {
mytitle <- paste0("1 - pnorm(", lo, ") = ",
round(1 - pnorm(lo), 4))
addtox <- lo
ydouble <- c(0,0)
arrows <- list(
annotate("segment", x = lo, y = -0.05, size = 0.8,
colour = "darkorchid",
xend = -4, yend = -0.05, arrow = arrow()),
annotate("text", x = mean(c(lo, -4)), y = -0.05, vjust = -0.5,
label = paste0("pnorm(", lo, ")=", round(pnorm(lo), 3)))
)
}
ggplot() +
geom_ribbon(aes(xmin = xrib, x=xrib, ymin = 0, ymax = yrib),
fill = "darkorchid", alpha = 0.4) +
geom_line(aes(x = xseq, y = yseq), linewidth = 1) +
scale_x_continuous(breaks = c(seq(-4,4,2))) +
theme_bw() +
annotate("text", x = addtox, y = rep(-0.01, length(addtox)),
label = addtox) +
labs(x = "x", y = "dnorm(x)", title = mytitle,
caption = "Created by Devan Becker\nGithub: DBecker7/DB7_TeachingApps") +
theme(title = element_text(size = 16),
axis.text = element_text(size = 14)) +
arrows
})
}
# Run the application
shinyApp(ui = ui, server = server)Verify the following with the Z-tables (and try them in the app above):
4.3 The Inverse CDF
d* series of functions.q
qnorm(p = p, mu = mu, sigma = sigma) calculates the value \(q\) such that \(P(X = q) = p\), where \(X \sim Norm(\mu, \sigma)\).