Exercise 10

Let U, V be random numbers chosen independently from the interval [0, 1]. Find the cumulative distribution and density for the random variables (a) Y = max(U, V ). (b) Y = min(U, V ).

#Setting the seed for reproducibility
set.seed(123)

#Simulating random variables U and V
U <- runif(1000, 0, 1)  # Generate 1000 random numbers from a uniform distribution on [0, 1]
V <- runif(1000, 0, 1)  # Generate another set of 1000 random numbers from a uniform distribution on [0, 1]

#(a) Y = max(U, V)
Y_max <- pmax(U, V)  # Compute the element-wise maximum of U and V

#(b) Y = min(U, V)
Y_min <- pmin(U, V)  # Compute the element-wise minimum of U and V

#Plotting histograms to visualize the distributions
par(mfrow = c(1, 2))  # Set up a 1x2 grid for side-by-side plots

hist(Y_max, main = "Distribution of max(U, V)", col = "lightblue", xlab = "Y_max", freq = FALSE)
#Plotting a histogram for Y_max, with a title, light blue color, and no frequency counts on the y-axis

hist(Y_min, main = "Distribution of min(U, V)", col = "lightgreen", xlab = "Y_min", freq = FALSE)

#Plotting a histogram for Y_min, with a title, light green color, and no frequency counts on the y-axis
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