Lecture 19: Joint, Conditional and Marginal Distributions; 2D LOTUS; Expected Distance between Uniforms; Chicken-egg Problem

Stat 110, Prof. Joe Blitzstein, Harvard University

Joint, Conditional and Marginal Distributions

Joint CDF

A joint CDF is simply where we are dealing with multiple random variables. As an example, a case where we have two random variables $X, Y$, the joint CDF of two random variables $X, Y$ can be expressed as:

\begin{align} F(x,y) &= P(X {\le} x, Y {\le} y) \end{align}

Note that the random variables may be discrete, continuous, or a mixture of both.

Joint PDF

The joint PDF, in the case of continuous random variables, is what you would integrate to get the joint CDF. Continuing with our example of two (continous) random variables, we have:

\begin{align} f(x,y) &= \frac{\partial^2}{\partial{x}\partial{y}} F(x,y) \end{align}

Conversely, if we want to know the probability of $X,Y$ in some set $A$, we integrate the density to get that probability.

\begin{align} P\left((X,Y) \in A\right) &= \iint\limits_{A} f(x,y) \, dxdy \end{align}

Integrate by holding one variable constant, and then do the other. The key thing is to be sure to get the limits of integration correct.

Marginal PDF

The marginal PDF of $X$ is obtained by integrating out the $Y$. Recall the $X,Y$ contigency table and the definition of marginal probability.

\begin{align} \int_{-\infty}^{\infty} f(x,y) \, dy \end{align}

Notice that by keeping $X$ constant and integrating over all $Y$, the marginal PDF of $X$ no longer depends on $Y$.

And we can do vice-versa for the marginal PDF of $Y$, but keeping $Y$ constant and integrating over all $X$.

Do not forget that taking the marginal PDF of $X$ and then integrating over all $X$, we should get 1.0.

\begin{align} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} f(x,y) \, dx dy &= 1.0 \end{align}

Conditional PDF

Given that we know $X$, what is the appropriate PDF for $Y$?

Well, we can apply what we know about conditional probability to get a conditional PDF.

\begin{align} f_{Y|X} (y|x) &= \frac{f_{XY}(x,y)}{f_{X}(x)} \\ &= \frac{f_{X|Y}(x|y) \, f_{Y}(y)}{f_{X}(x)} \end{align}

This is completely analogous to conditional probability.


$X,Y$ are independent if

\begin{align} f_{X,Y}(x,y) &= f_{X}(x) \, f_{Y}(y) &\quad \text{for all }x, y \text{ from PDF p.o.v.} \\ \\ F(x,y) &= F(x) \, F(y) &\quad \text{for all }x, y \text{ from CDF p.o.v.} \end{align}

These statements are equivalent, but in most cases it might be easier to work the PDFs.

A Uniform example

Let's revisit that distribution that is uniform on the unit disc $x^2 + y^2 \le 1$.

In [1]:
import matplotlib
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline  


fig = plt.figure(figsize = (5.0,5.0))
ax = fig.add_subplot(1, 1, 1)
circ = plt.Circle((0, 0), radius=1.0, color="b", alpha=0.2, lw=5)
plt.axhline(0, color="black", alpha=0.4)
plt.axvline(0, color="black", alpha=0.4)

Joint PDF

A valid PDF is required to integrate to 1.0, and since we are looking at a Uniform distribution over the unit circle with area equal to $\pi$, we have

\begin{align} f_{XY}(x,y) &= \begin{cases} \frac{1}{\pi} & \quad \text{if } x^2 + y^2 \le 1 \\ 0 & \quad \text{otherwise}\\ \end{cases} \end{align}

Understand that while this is uniformly distributed, $x$ and $y$ are closely related with $x^2 + y^2 \le 1$, and so $X,Y$ cannot be independent. More on that coming up.

Marginal PDF of $X$

\begin{align} f_{X}(x) &= \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \frac{1}{\pi} dy \\ \\ &= \frac{2}{\pi} \sqrt{1-x^2} &\quad \text{where } -1 \le x \le 1 \end{align}

Notice that while the joint PDF is Uniform, the marginal PDF is not Uniform: it grows as $x$ approaches 0.

By symmetry we can just replace $x$ with $y$ to get the marginal PDF of $Y$ $f_{Y}(y)$

Conditional PDF of $Y|X$

\begin{align} f_{Y|X}(y|x) &= \frac{f_{XY}(x,y)}{f_{X}(x)} \\ &= \frac{\frac{1}{\pi}}{\frac{2}{\pi} \sqrt{1-x^2}} \\ &= \frac{1}{2 \sqrt{1-x^2}} &\quad \text{if } -\sqrt{1-x^2} \le y \le \sqrt{1-x^2} \end{align}

But since we are treating $x$ as a constant, and the above equation does not depend on $y$, the conditional PDF $f_{Y|X}(y|x)$ is actually Uniform.

\begin{align} Y|X &\sim \operatorname{Unif}(-\sqrt{1-x^2}, \sqrt{1-x^2}) &\quad \text{or also } \\ \\ Y|X=x \, &\sim \operatorname{Unif}(-\sqrt{1-x^2}, \sqrt{1-x^2}) \end{align}


Since in our example above

\begin{align} f_{XY}(x,y) \neq f_{X}(x) \, f_{Y}(y) \end{align}

we can see that $X,Y$ are not independent.

Or, because the unconditional distribution of $Y$ is not the same as the conditional distribution of $Y|X$, it necessarily follows that $X,Y$ are not independent as learning $X$ gives us information about $Y$ (and vice-versa).


Let $X,Y$ have joint PDF $f(x,y)$, and let $g(x,y)$ be a real-valued function of $x,y$.

Then, without trying to find the PDF of $g(x,y)$, LOTUS lets us use joint PDF:

\begin{align} \mathbb{E} g(x,y) &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(x,y) \, f(x,y) \, dx dy \end{align}

We will illustrate this by proving the following theorem.


If $X,Y$ are independent, then $\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$.

Or, in other words, the independence of r.v. $X$ and $Y$ implies that they are uncorrelated.


Consider the continuous case

\begin{align} \mathbb{E}(XY) &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} xy \, f(x,y) \, dx dy &\quad \text{2D LOTUS} \\ &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} xy \, f_{X}(x) \, f_{Y}(y) \, dx dy &\quad \text{since } X,Y \text{ are independent} \\ &= \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} xy \, f_{X}(x) \, f_{Y}(y) \, dx \right) dy &\quad \text{this is a double integral, so we can group thusly} \\ &= \int_{-\infty}^{\infty} y \, f_{Y}(y) \underbrace{ \left( \int_{-\infty}^{\infty} x \, f_{X}(x) \, dx \right)}_{\text{actually }\mathbb{E}(X) \text{, which is constant}} dy &\quad \text{since we are holding } y \text{ constant} \\ &= \mathbb{E}(X) \int_{-\infty}^{\infty} y \, f_{Y}(y) \, dy \\ &= \mathbb{E}(X) \, \mathbb{E}(Y) &\quad \blacksquare \end{align}

Example: expected distance between 2 Uniformly distributed random points

Given i.i.d. $X,Y \sim Unif(0,1)$, find the expected distance between $X$ and $Y$, $\mathbb{E}|X-Y|$.

In [2]:

fig,ax = plt.subplots(1, figsize=(6.0,1.0))

plt.plot([1/3, 2/3],[0,0], 'ro')
ax.axhline(y=0, color='k', alpha=0.2)

ax.set_xticks([0.0, (1.0/3.0), (2.0/3.0), 1.0])
ax.set_xticklabels(['0', '1/3', '2/3', '1'])



We don't need to directly find the distribution of $|X-Y|$, because we are only interested in the average distance between points $x,y$. Using LOTUS, we have:

\begin{align} \mathbb{E}|X-Y| &= \int_{0}^{1} \int_{0}^{1} |x-y| \, dx dy &\quad \text{since }X,Y \text{ are i.i.d., the joint PDF }=1 \\ &= \iint_\limits{x \gt y} (x-y) \, dx dy + \iint_\limits{x \le y} (y-x) \, dx dy &\quad \text{split into 2 integrals to deal with abs. value} \\ &= 2 \int_{0}^{1} \int_{y}^{1} (x-y) \, dx dy &\quad \text{by symmetry; and since } x \gt y \text{, inner integral starts from }y \\ &= 2 \int_{0}^{1} \left( \frac{x^2}{2} - yx \right) \bigg|_{y}^{1} \, dy \\ &= 2 \int_{0}^{1} \frac{1}{2} - y + \frac{y^2}{2} \, dy \\ &= 2 \left( \frac{1}{6} \right) \\ &= \boxed{ \frac{1}{3} } \end{align}

But looking at that line-segment illustration above suggests another way of looking at this problem.

Let $M = max(X,Y)$

Let $L = min(X,Y)$

And so

\begin{align} |X-Y| &= M - L \\ \\ \mathbb{E}(M-L) &= \frac{1}{3} &\quad \text{from our previous proof} \\ \mathbb{E}(M) - \mathbb{E}(L) &= \frac{1}{3} &\quad \text{since } X,Y \text{ are i.i.d.} \\ \\ \mathbb{E}(M+L)&= \mathbb{E}(M) + \mathbb{E}(L) &\quad \text{by linearity} \\ &= \frac{1}{2} + \frac{1}{2} &\quad \text{since} X \sim \operatorname{Unif}(0,1) \text{, } Y \sim \operatorname{Unif}(0,1) \\ &= 1 \\ \\ \mathbb{E}(M) &= \frac{2}{3} \\ \mathbb{E}(L) &= \frac{1}{3} \end{align}

Chicken-egg Problem

Say we have $N$ eggs; let $N \sim \operatorname{Pois}(\lambda)$

Some of the eggs hatch, while some don't. Each egg hatches with probability $p$. The event of an egg hatching is independent of the others.

Let $X$ be the number of eggs out of $N$ that do hatch, so $X|N \sim \operatorname{Bin}(N,p)$.

Let $Y$ be the number of eggs that don't hatch, so $X+Y = N$.

Find the joint PMF of $X,Y$. Are they independent?

\begin{align} P(X{=}i, Y{=}j) &= \sum_{n=0}^{\infty} P(X{=}i, Y{=}j |N{=}n) \, P(N{=}n) \\ \\ &= P(X{=}i, Y{=}j |N{=i+j}) \, P(N{=i+j}) &\quad \text{since we know that } i + j = n \\ \\ &= P(X{=}i | N{=i+j}) \, P(N{=i+j}) &\quad Y{=}j \text{ is redundant} \\ \\ &= \binom{i+j}{i} p^i \, q^j \cdot \frac{e^{-\lambda} \lambda^{i+j}}{(i+j)!} &\quad \text{from definition of binomial and Poisson} \\ \\ &= \frac{(i+j)!}{i! \, j!} p^i \, q^j \cdot \frac{e^{-\lambda} \lambda^{i+j}}{(i+j)!} \\ \\ &= \frac{(\lambda p)^i}{i!} \, \frac{(\lambda q)^j}{j!} \, e^{-\lambda} \\ \\ &= \left(e^{-\lambda p} \frac{(\lambda p)^i}{i!} \right) \left(e^{-\lambda q} \frac{(\lambda q)^j}{j!} \right) &\quad \text{since } p + q = 1 \\ \\ &\Rightarrow X,Y \text{ are independent, } X \sim \operatorname{Pois}(\lambda p), Y \sim \operatorname{Pois}(\lambda q) \end{align}