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)

Note that the function is defined or all \(x, y > 0\), but most of the density is in this region around the origin.

Computing Joint Probabilities

We represent the following probability graphically:

\[ P(X > 1, Y > .5) = \int_{y = .5}^{\infty} \int_{x = 1}^{\infty} f(x, y) dx, dy \]

z2 <- z
z2[1:10,] <- 0
z2[, 1:5] <- 0

persp(x, y, z2, theta = 25, phi = 25, zlim = c(0, max(z)))

Computing the probability analytically

On board.

Computing the probability computationally

We approximate the integral by discretizing the continuous volume into a series of rectangles of height \(f(x, y)\) and base area \(dxdy\) and summing the probabilities.

\[ P(X > 1, Y > .5) \approx \sum_{x,y\textrm{ in grid}} f(x, y) dx, dy \]

sum(z2 * dx * dy)

## [1] 0.03797101

Practice

We can visualize \(P(X > 1 \textrm{ or } Y > .5)\),

z3 <- z
z3[1:10, 1:5] <- 0
persp(x, y, z3, theta = 25, phi = 25, zlim = c(0, max(z)))

then approximate it computationally.

sum(z3 * dx * dy)

## [1] 0.4165207

Or we can visualize the complement, \(P(X < 1 \textrm{ or } Y < .5)\),

z4 <- z
z4[11:31, ] <- 0
z4[, 6:31] <- 0
persp(x, y, z4, theta = 25, phi = 25, zlim = c(0, max(z)))

and compute 1 minus its volume.

1 - sum(z4 * dx * dy)

## [1] 0.1421356

The fact that these methods aren’t in close agreement is likely due to the disagreement between the discrete and continuous densities around the origin.

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)