Joint Density Functions

We can visualize part of the following joint density function using as a 3D surface.

\[ f(x, y) = 6e^{-2x- 3y} \quad \quad (x, y > 0) \]

f <- function(x, y) {
  6 * exp(-2 * x - 3 * y)
}

dx <- .1
dy <- .1
x <- seq(0, 3, by = dx)
y <- seq(0, 3, by = dy)
z <- outer(x, y, f)
persp(x, y, z, theta = 25, phi = 25)

Marginal Probabilities

We can visualize the process of integrating out \(x\) to get \(f_Y(y)\).

fy <- colSums(z)
plot(y, fy, pch = 16)
lines(y, fy, col = "tomato", lwd = 2)

And equivalently for \(f_X(x)\)

fx <- rowSums(z)
plot(y, fy, pch = 16)
lines(y, fy, col = "tomato", lwd = 2)
points(x, fx, pch = 16)
lines(x, fx, col = "steelblue", lwd = 2)

Cumulative Density Functions

For our same joint pdf, we can consider the joint cdf.

\[ F(x, y) = \int_{s = 0}^x \int_{t = 0}^y 6 e^{-2s - 3t} dt ds \] This function has the following shape:

Fxy <- function(x, y) {
  exp(-3*y - 2*x) - exp(-2*x) - exp(-3*y) + 1
}
w <- outer(x, y, Fxy)
persp(x, y, w, theta = 25, phi = 25)

Relatively constrained probabilities

We wish to compute the following probability.

\[ P(X > 2Y) = P(Y < \frac{1}{2}X) \] We can visualize the set over which we wish to integrate,

zconstrained <- z
for(i in 1:nrow(zconstrained)) {
  for(j in floor(.5*i):ncol(zconstrained)){
    zconstrained[i, j] <- 0
  }
}
persp(x, y, zconstrained, theta = 25, phi = 25)

and we can approximate the volume using the discrete sum.

sum(zconstrained * dx * dy)

## [1] 0.3582753

Or alternatively by taking the mean density of the points on the grid times the area under integration.

mean(zconstrained[zconstrained > 0])*9/4

## [1] 0.3838664

Monte Carlo Integration (Accept-Reject)

n <- 1e6
x <- rexp(n, rate = 2)
y <- rexp(n, rate = 3)
mean(y<.5*x)

## [1] 0.428594

Correlation/Covariance

Because \(X\) is independent of \(Y\), we know that the covariance and correlation are zero. If we simulate from \(f(x,y)\), we can compute this approximately.

cov(x, y)

## [1] 0.000187333

cor(x, y)

## [1] 0.001122505