Marginal Distribution (Continuous)

Deep Breakdown

What it tells me (Situation Awareness): For a continuous bivariate distribution, the marginal PDF of X is obtained by integrating the joint PDF over all possible values of Y. This "integrates Y out", leaving only the behaviour of X. It is the continuous analogue of summing the rows of a discrete joint table. In DA/DS this is essential when working with multivariate Gaussian models, copulas, and Bayesian inference where you compute $p(\theta) = \int p(\theta, z) \, dz$ to get the marginal posterior.

Flowchart

Real-world use

Where it is used

You fit a bivariate distribution to two continuous features (e.g., age and spending) and want to analyse each feature's distribution individually, without conditioning on the other.

Problem it solves

After fitting a joint model, you often need the distribution of just one variable for reporting, simulation, or comparison. The marginal PDF extracts this without requiring a new fit.

How it solves the problem

Integrating over all Y values collapses the joint density into a univariate density. The resulting $f_X(x)$ is a valid PDF that sums (integrates) to 1 and fully describes X's distribution on its own.

Example in a business

A DS fits a bivariate Gaussian to (session_duration, revenue). The marginal PDF of revenue is then used to set revenue threshold alerts and compute the probability that any given user session generates revenue above a target, marginalising over session duration.

The formula, piece by piece

Full formula: $f_X(x) = \int_{-\infty}^\infty f(x,y) \, dy$

The formula in words

"The marginal density of X at value x equals the integral of the joint density over all possible y values, with x held fixed." You are adding up all the joint density along the vertical slice at position x.

Critical Concepts

• Integrating out: the Y variable disappears after integration; only x remains.
• Valid PDF: $f_X(x) \ge 0$ and $\int f_X(x) \, dx = 1$.
• Information loss: marginals do not recover the joint; the dependency structure is lost.
• Bivariate Gaussian marginals: each marginal of a bivariate Gaussian is itself Gaussian.
• Numerical integration: in practice, use scipy.integrate.quad or Monte Carlo integration when analytical form is unavailable.

Step-by-step calculation (tiny example)

Joint PDF: $f(x,y) = x + y$ for $0 \le x \le 1$, $0 \le y \le 1$. Find marginal $f_X(x)$.

Step 1: Set up the integral over y $f_X(x) = \int_0^1 (x + y) \, dy$

Step 2: Integrate $= [xy + y^2/2]_0^1 = x + 1/2$

Step 3: Result $f_X(x) = x + 1/2$ for $0 \le x \le 1$

Step 4: Verify normalisation $\int_0^1 (x + 1/2) \, dx = [x^2/2 + x/2]_0^1 = 1/2 + 1/2 = 1.0$

Code examples

Beginner example (Python)

import numpy as np
from scipy import integrate


# Joint PDF: f(x,y) = x + y on [0,1] x [0,1]
def joint_pdf(y: float, x: float) -> float:
    return x + y


# Compute marginal f_X(x) numerically for several x values
x_grid = np.linspace(0, 1, 6)
for x_val in x_grid:
    marginal, _ = integrate.quad(joint_pdf, 0, 1, args=(x_val,))
    # Analytical: f_X(x) = x + 0.5
    analytical = x_val + 0.5
    print(f"x={x_val:.2f}: f_X(x) = {marginal:.4f} (analytical: {analytical:.4f})")

Beginner example (R)

# Marginal f_X(x) = integrate f(x,y) dy from 0 to 1
joint_pdf <- function(y, x_val) x_val + y
x_grid <- seq(0, 1, by = 0.2)
for (x_val in x_grid) {
  result <- integrate(joint_pdf, lower = 0, upper = 1, x_val = x_val)
  cat(sprintf("x=%.1f: f_X(x) = %.4f (analytical: %.4f)\n", x_val, result$value, x_val + 0.5))
}

Professional example (Python)

import numpy as np
from scipy import integrate
from scipy.stats import multivariate_normal


def bivariate_gaussian_marginal_x(
    x_grid: np.ndarray,
    mean: list[float],
    cov: list[list[float]],
    y_lower: float = -10.0,
    y_upper: float = 10.0,
) -> np.ndarray:
    # Compute marginal PDF of X by numerically integrating the bivariate Gaussian.
    # For bivariate Gaussian, this recovers N(mean_x, sigma_x^2) analytically.
    dist = multivariate_normal(mean=mean, cov=cov)
    marginal_densities = np.zeros_like(x_grid, dtype=float)
    for i, x_val in enumerate(x_grid):

        def integrand(y: float) -> float:
            return dist.pdf([x_val, y])

        marginal_densities[i], _ = integrate.quad(integrand, y_lower, y_upper)
    return marginal_densities


if __name__ == "__main__":
    from scipy.stats import norm
    MEAN = [5.0, 3.0]
    COV = [[4.0, 1.5], [1.5, 2.0]]
    x_vals = np.linspace(0, 10, 50)
    numerical_marginal = bivariate_gaussian_marginal_x(x_vals, MEAN, COV)
    analytical_marginal = norm.pdf(x_vals, loc=MEAN[0], scale=COV[0][0] ** 0.5)
    max_error = np.max(np.abs(numerical_marginal - analytical_marginal))
    print(f"Max absolute error (numerical vs analytical): {max_error:.2e}")
    print("Numerical integration recovers the Gaussian marginal correctly.")

Source: AIO p. 85`

#probability-statistics#multivariable-statistics-and-information#joint-and-marginal-distributions