true

Question 1

#a
lambda <- 4.5
p_a <- ppois(7, lambda)
p_a
## [1] 0.9134135
#b
p_b <- ppois(5, lambda) - ppois(1, lambda)
p_b
## [1] 0.641831

Question 2

#a
mu <- 5
sigma <- 3
p2_a <- 1-pnorm(7.8, mean = mu, sd = sigma)
p2_a
## [1] 0.1753239
#b
p2_b <- pnorm(7.8, mean = mu, sd = sigma) - pnorm(1.6, mean = mu, sd = sigma)
p2_b
## [1] 0.6961389
#c
y_95 <- qnorm(0.95, mean = mu, sd = sigma)
y_95
## [1] 9.934561

Question 3

rate <- 9
curve(dexp(x, rate = rate), 
      from = 0, to = 1, 
      main = expression("Exponential PDF (" * lambda == 9 * ")"), 
      xlab = "x", 
      ylab = "Density", 
      col = "blue", 
      lwd = 2)

Question 4

set.seed(42)
x <- rnorm(10000, mean = 1, sd = 2)
sample_mean <- mean(x)
sample_sd <- sd(x)
c(Sample_Mean = sample_mean, Sample_SD = sample_sd)
## Sample_Mean   Sample_SD 
##   0.9773811   2.0122696
cat("Theoretical Mean =", 1, "\nTheoretical SD =", 2, "\n")
## Theoretical Mean = 1 
## Theoretical SD = 2
cat("Difference in Mean =", sample_mean - 1, "\nDifference in SD =", sample_sd - 2)
## Difference in Mean = -0.0226189 
## Difference in SD = 0.01226965