1. Demonstrate PDF, CDF and Quantile plots for the standard normal
distribution
library(ggplot2)
PDF and CDF plots for the Standard Normal Distribution
Generate the x values
x <- seq(-5, 5, length.out = 10)
x
## [1] -5.0000000 -3.8888889 -2.7777778 -1.6666667 -0.5555556 0.5555556
## [7] 1.6666667 2.7777778 3.8888889 5.0000000
Generate PDF and CDF values
pdf_values <- dnorm(x)
pdf_values
## [1] 1.486720e-06 2.074403e-04 8.421534e-03 9.947714e-02 3.418923e-01
## [6] 3.418923e-01 9.947714e-02 8.421534e-03 2.074403e-04 1.486720e-06
cdf_values <- pnorm(x)
cdf_values
## [1] 2.866516e-07 5.035210e-05 2.736602e-03 4.779035e-02 2.892574e-01
## [6] 7.107426e-01 9.522096e-01 9.972634e-01 9.999496e-01 9.999997e-01
Generate PDF and CDF plots
pdf_plot <- ggplot(data.frame(x, pdf_values), aes(x = x, y = pdf_values)) +
geom_line(color = "red") +
labs(title = "PDF of Standard Normal Distribution",
x = "x values",
y = "PDF values")
pdf_plot
cdf_plot <- ggplot(data.frame(x, cdf_values), aes(x = x, y = cdf_values)) +
geom_line(color = "blue") +
labs(title = "CDF of Standard Normal Distribution",
x = "x values",
y = "CDF values")
cdf_plot
Quantile plot for Standard Normal Distribution
Generate probabilities for Quantile values
p <- seq(0.1, 0.9, length.out = 10)
Generate Quantile values
quantile_values <- qnorm(p)
quantile_values
## [1] -1.2815516 -0.8819982 -0.5894558 -0.3406948 -0.1116372 0.1116372
## [7] 0.3406948 0.5894558 0.8819982 1.2815516
Generate Quantile plot for Standard Normal Distribution
quantile_plot <- ggplot(data.frame(p, quantile_values), aes(x = p, y = quantile_values)) +
geom_line(color = "green") +
labs(title = "Quantile plot for Standard Normal Distribution",
x = " Probabilities",
y = "Quantile values")
quantile_plot
Generate PDF, CDF, and Quantile plots for Standard Normal
Distribution
install.packages("gridExtra", repos = "https://cloud.r-project.org/")
## Installing package into 'C:/Users/alxn0/AppData/Local/R/win-library/4.4'
## (as 'lib' is unspecified)
## package 'gridExtra' successfully unpacked and MD5 sums checked
##
## The downloaded binary packages are in
## C:\Users\alxn0\AppData\Local\Temp\RtmpKGn94V\downloaded_packages
library(gridExtra)
grid.arrange(pdf_plot, cdf_plot, quantile_plot, nrow = 1)
2. Create a function in R to calculate and draw CDF from the stated
PDF
pdf_function <- function(x) {
ifelse(x >= 0 & x < 3, 0.2,
ifelse(x >= 3 & x < 5, 0.05,
ifelse(x >= 5 & x < 6, 0.15,
ifelse(x >= 7 & x < 10, 0.05, 0))))
}
cdf <- function(x, pdf_function) {
cdf_values <- numeric(length(x))
for (i in seq_along(x)) {
cdf_values[i] <- integrate(pdf_function, lower = 0, upper = x[i]) $value
}
return(cdf_values)
}
cdf_values
## [1] 2.866516e-07 5.035210e-05 2.736602e-03 4.779035e-02 2.892574e-01
## [6] 7.107426e-01 9.522096e-01 9.972634e-01 9.999496e-01 9.999997e-01
Generate x values
x <- seq(0, 10, length.out = 100)
Compute PDF and CDF values
pdf_values <- sapply(x, pdf_function)
cdf_values <- cdf(x, pdf_function)
Generate data frame
df <- data.frame(x = x, PDF = pdf_values, CDF = cdf_values)
Generate PDF and CDF plots
ggplot(df, aes(x)) +
geom_line(aes(y = PDF, color = "PDF")) +
geom_line(aes(y = CDF, color = "CDF")) +
labs(title = "PDF and CDF Plots",
x= "x value",
y = "Function") +
scale_color_manual(values = c("PDF" = "blue", "CDF" = "purple"))
3. Create a function to calculate posterior probability using Bayes
theorem.
bayes_posterior <- function(prior, likelihood, marginal_evidence) {
posterior <- (likelihood * prior) / marginal_evidence
return(posterior)
}
4. Bayes Theorem Question
A certain virus infects one in every 400 people. A test used to
detect the virus in a person is positive 85% of the time if the person
has the virus and 5% of the time if the person does not have the virus.
(This 5% result is called a false positive). Let A be the event “the
person has the virus” and B be the event “the person tests
positive”.
a) Find the probability that a person has the virus given that they
have tested positive, i.e. find P(A|B). Round your answer to the nearest
hundredth of a percent.
b) Find the probability that a person does not have the virus given
that they test negative, i.e. find P(A’|B’). Round your answer to the
nearest hundredth of a percent.
# Function to compute posterior probability using Bayes' Theorem
bayes_posterior <- function(prior, likelihood, marginal_evidence) {
posterior <- (likelihood * prior) / marginal_evidence
return(posterior)
}
# Function to calculate both probabilities
virus_test_probabilities <- function(prevalence, sensitivity, false_positive_rate) {
# Given probabilities
prior <- prevalence # P(A)
prior_neg <- 1 - prior # P(A')
likelihood <- sensitivity # P(B | A)
false_neg_rate <- 1 - sensitivity # P(B' | A)
false_pos_rate <- false_positive_rate # P(B | A')
specificity <- 1 - false_pos_rate # P(B' | A')
# Compute P(B)
marginal_evidence_B <- (likelihood * prior) + (false_pos_rate * prior_neg)
# Compute P(A | B) - Probability of having the virus given a positive test
P_A_given_B <- bayes_posterior(prior, likelihood, marginal_evidence_B)
# Compute P(B')
marginal_evidence_B_neg <- (false_neg_rate * prior) + (specificity * prior_neg)
# Compute P(A' | B') - Probability of NOT having the virus given a negative test
P_A_neg_given_B_neg <- bayes_posterior(prior_neg, specificity, marginal_evidence_B_neg)
# Return both probabilities
return(list(P_A_given_B = round(P_A_given_B * 100, 2),
P_A_neg_given_B_neg = round(P_A_neg_given_B_neg * 100, 2)))
}
# Example Usage:
result <- virus_test_probabilities(prevalence = 1/400, sensitivity = 0.85, false_positive_rate = 0.05)
# Print results
print(paste("P(A | B) =", result$P_A_given_B, "%"))
## [1] "P(A | B) = 4.09 %"
print(paste("P(A' | B') =", result$P_A_neg_given_B_neg, "%"))
## [1] "P(A' | B') = 99.96 %"