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 $F : R \to [0, 1]$ is defined $F (x) = P (X \leq x)$ .

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 : R \to R

f (x) = \frac{d}{d x} 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) d x = 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: $\begin{aligned} P (X > x) & = 1 - P (X \leq x) = 1 - F (x) \\ P (x_{1} < X \leq x_{2}) & = P (X \leq x_{2}) - P (X \leq x_{1}) = F (x_{2}) - F (x_{1}) \end{aligned}$ For the CDFs of continuous random variables,

You can assume the CDF is differentiable.
$P (X \leq x) = P (X < x)$ , so you can swap out $\leq$ 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: $\begin{aligned} P (X \leq x) = P (X < x) & = \int_{- \infty}^{x} f (x) d x \\ P (X \geq x) = P (X > x) & = \int_{x}^{\infty} f (x) d x \\ P (x_{1} < X < x_{2}) & = \int_{x_{1}}^{x_{2}} f (x) d x . \end{aligned}$

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	$E (X) = \sum_{x} x P (X = x)$	$E (X) = \int_{- \infty}^{\infty} x f_{X} (x) d x$
LOTUS	$E (g (X)) = \sum_{x} g (x) P (X = x)$	$E (g (X)) = \int_{- \infty}^{\infty} g (x) f_{X} (x) d x$
Bayes’ rule	$P (X = x \| Y = y) = \frac{P (Y = y \| X = x) P (X = x)}{P (A)}$	$f_{X \| Y = y} (x) = \frac{f_{Y \| X = x} (y) f_{X} (x)}{f_{Y} (y)}$

2. Uniform

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