MATH 180A Lecture 8

Distributions

Probability Distribution

Let () be a probability space and a random variable. The (probability) distribution of is the probability measure on defined by

is the sample space

Examples.

Fair coin toss: $\Omega =\{H,T\}$. Define $X:\Omega \to \mathbb{R}$ by $X(H)=\pi ,X(T)=-\sqrt{2}$. What is the distribution of $X$?

Roll a fair die twice: $\Omega =\{1,2,3,4,5,6{\}}^2$. Define $S:\Omega \to \mathbb{R}$ by $S((i,j))=i+j$. What is the distribution of $S$?

e.g.

Choose a point uniformly at random on a random disk: $\Omega =\{(x,y)\in {\mathbb{R}}^2:x^2+y^2\le 1\}$. Define $X:\Omega \to \mathbb{R}$ by $X((x,y))=\sqrt{x^2+y^2}$. What is the distribution of $X$?

We’ll learn a good way to describe this distribution in 3.1-3.2.

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Sections 3.1 and 3.2

Cumulative Distribution Functions

Cumulative Distribution Function (CDF)

Let be a random variable. The cumulative distribution function (cdf) of is the function defined by

Crucial Fact: The cdf completely determines the distribution. Meaning, if , then .

For example, suppose . What is ?

.

Examples.

A fair coin is tossed 3 times. Let $X$ = "number of heads". Find the cdf of $X$ and sketch its graph.

A point is chosen uniformly at random from the unit disk. Let $X$ = "distance of the point from the center". Find the cdf of $X$ and sketch its graph.