MATH 180A Lecture 9

Discrete & Continuous Random Variables, Probability Mass & Density Functions

There are two important types of probability measure that we will focus on.

Discrete, Probability Mass Function (pmf)

A probability measure on is discrete if there exists a finite or infinite sequence of numbers such that

A random variable is discrete if its distribution is discrete.
In this case, we call the function defined by the probability mass function of (pmf)

Crucial fact: pmf completely determines the distribution of a discrete random variable. Because .

Continuous, Probability Density Function (pdf)

A probability measure on is continuous if there exists an integrable function s.t.

A random variable is continuous if its distribution is continuous.
In this case, we call the probability density function (pdf) of .

Note

Crucial fact: pdf completely determines the distribution of continuous random variables.

Examples.

A fair coin is tossed 3 times. Let $X$ = "number of heads". $X$ is a discrete random variable. Find its pmf.

for all other values of

A point is chosen uniformly at random from the unit disk. Let $X$ = "distance of the point from the cetner". $X$ is a continuous random variable. Find its pdf.

take the derivative of each side
Fundamental theorem of calculus

Uniform

We say that a random variable is uniform on [a,b] if is continuous and its pdf is given by

In this case we write .

Example. Suppose . Calculate .

In general,

Going from pmf/pdf to cdf

The formulas to go from the pmf/pdf of a discrete/continuous random variable to its cdf are straightforward:

• If is discrete, .
• If is continuous, .

Examples.

Suppose $X$ is discrete w/ pmf $p_X(-\pi )=\frac{1}{3},p_X(\sqrt{2})=\frac{2}{3}$. Find its cdf.

Suppose $X\sim \text{Unif}[-1,2]$. Find its cdf.

Going from cdf to pmf/pdf

The formulas to go from the cdf to the pmf/pdf of a discrete/continuous random variable are slightly more involved.

Fact: Let be a random variable and its cdf.

• If is piecewise constant, then is discrete and its pmf is found by calculating the magnitude of the jumps at the points on discontinuity.
• If is continuous and piecewise continuously differentiable, then is continuous and its pdf is given by .

Examples.

Suppose $X$ has cdf given by $F_X(s)=\{\begin{array}{ll}0& s<-\pi \\ \frac{1}{3}& -\pi \le s<\sqrt{2}\\ 1& \sqrt{2}\le s\end{array}$. Is $X$ discrete or continuous? If so, find its pmf/pdf.

Suppose $X$ has cdf given by $F_X(s)=\{\begin{array}{ll}0& s<0\\ \sin s& 0\le s<\frac{\pi }{2}\\ 1& \frac{\pi }{2}\le s\end{array}$. Is $X$ discrete or continuous? If so, find its pmf/pdf.

is continuous + continuously piecewise differentiable is continuous and

Defining Properties of pdf’s, pmf’s, pdf’s

Fact

A function is the cdf of a random variable if and only if

1. and
2. Nondecreasing meaning
3. Right-continuous comes from right-continuous
   not left-continuous

Fact

A function is the pmf of a discrete random variable if and only if there exists a finite or infinite sequence of numbers such that

1. for all
3. ,

Fact

A function is the pdf of a continuous random variable if and only if is integrable and