Ch 3: Random Variables and Probability Distributions

Random Variable

A function that associates a real numbers with outcomes in sample space

Probability Distribution Criteria (PDF)

1. f(x) is postive

2. Adding all outcomes’ probabilities = 1

3. P(X = x) = f(x)

What does P(X=x) imply?

f(x). For example, P(X=3) will tell you the probability outcome of 3.

PDF notation

f(x)

Cumulative Distribution Function (CDF)

Looks for a range of values leading up to a certain value

PDF vs. CDF

-PDF: Looking for an exact value of X

-CDF: Looking for a range of values less than or equal to a certain value

Joint Probability Distribution Criteria

1. f(x, y) is positive for all (x,y)

2. Adding all f(x, y) = 1

3. P(X = x, Y = y) = f(x, y)

Marginal Distribution for X

-Integrate y out

-Use the bounds of y

Marginal Distribution for Y

-Integrate y out

-Use the bounds of y

Which integral to do first for a Joint Probability Distribution?

The first integral should have bounds containing the other variable (so second integral can be constant)

Conditional Distribution

The given random variable goes at bottom

Statistical Independence

Conditional * condition = f(x, y)

Why are random variables and probability distributions important?

-allows us to use historical/previously known data to determine likelihood of random outcome

-can be used to describe data and infer how future ones will fare