Week 5: Continuous Distributions

0. Logistical Info

Section date: 10/18
Associated lecture: 10/12
Associated pset: Pset 5, due 10/20
Office hours on 10/18 from 7-9pm at Quincy Dining Hall
Remember to fill out the attendance form

0.1 Summary + Practice Problem PDFs

1. Continuous Random Variables

A continuous random variable has an interval for its support.

More precisely: a continuous random variable has an uncountable support, while discrete random variables have finite/countably infinite supports.

We heavily lean on the cumulative distribution function (CDF): for any random variable $X$, the CDF $F: \mathbb R \to [0, 1]$ is defined $F(x) = P(X \le x)$.

We don’t use the probability mass function (PMF) anymore, because $P(X = x) = 0$ for any $x$. This probability $0$ does not mean “impossible,” though. We instead use the probability density function:

The probability density function of a random variable $X$ with CDF $F$ is a function $f: \mathbb{R} \to \mathbb{R}$: $$ f(x) = \frac{d}{dx} F(x) $$ Probability densities are the continuous analog to probabilities, but they are NOT probabilities. A valid PDF is nonnegative and satisfies $$ \int_{-\infty}^\infty f(x) dx = 1 $$

1.1 Uses of CDFs and PDFs

For any random variable $X$ (continuous or discrete), you can use the CDF to calculate the following: \begin{align*} P(X > x) &= 1 - P(X \le x) = 1-F(x)\\ P(x_1 < X \le x_2) &= P(X \le x_2) - P(X \le x_1) = F(x_2) - F(x_1) \end{align*} For the CDFs of continuous random variables,

You can assume the CDF is differentiable.
$P(X \le x) = P(X < x)$, so you can swap out $\le$ and $<$ in calculations like the above.

For a continuous random variable, we can find the probabilities of intervals by integrating the PDF and adjusting the bounds: \begin{align*} P(X \le x) = P(X < x) &= \int_{-\infty}^x f(x) dx\\ P(X \ge x) = P(X > x) &= \int_{x}^{\infty} f(x) dx\\ P(x_1 < X < x_2) &= \int_{x_1}^{x_2} f(x) dx. \end{align*}

1.2 Continuous analogs of all of our tools

The general rules are:

Integrals instead of sums
PDFs instead of PMFs

So here’s a table with the tools we’ve talked about:

Tool	Discrete	Continuous
Expectation	$E(X) = \sum_{x} x P(X = x)$	$E(X) = \int_{-\infty}^{\infty} x f_X(x) dx$
LOTUS	$E(g(X)) = \sum_x g(x) P(X = x)$	$E(g(X)) = \int_{-\infty}^\infty g(x) f_X(x) dx$
Bayes’ rule	$P(X = x \vert Y = y) = \frac{P(Y=y \vert X = x) P(X = x)}{P(A)}$	$f_{X\vert Y=y}(x) = \frac{f_{Y\vert X=x}(y) f_X(x)}{f_Y(y)}$

2. Uniform

For any interval $(a, b)$, we can define a uniform distribution with that support, denoted $U \sim \mathrm{Unif}(a, b)$. A uniform distribution is equivalent to having a constant PDF over the support. There is no uniform whose support is the full real line. \begin{align*} f_U(x) &= \begin{cases} \frac{1}{b-a} & x \in (a, b)\\ 0 & x \notin (a, b) \end{cases}\\ F_U(x) = P(U < x) &= \begin{cases} 0 & x \le a\\\ \frac{x-a}{b-a} & x \in (a, b)\\ 1 & x \ge b \end{cases} \end{align*}