Today

• Today: More of Chapter 2

• Reading: – Assignment #2 is up on the web site

– www.stat.lsa.umich.edu/~dbingham/stat405

– Please read Chapter 2

– Suggested problems: 2.4, 2.5, 2.7, 2.13, 2.25, 2.28, 2.32, 2R1, 2R2

Random Variables

• Often interested in a characteristic that varies from one individual to another

• A characteristic of the outcome of a random experiment is called a random variable (RV)

• RV’s are numeric or categorical

• Have discrete and continuous RV’s

• Usually use capital letters to denote an RV and small letters to denote outcomes

Random Variables

• More formally:

– For a given model, with sample space, Ω , a random variable, X(w), is a function from the sample space to the real numbers (where w is in the sample space)

• Each possible value, x, of a random variable, X, is an event (collection of outcomes from the sample space)

•

xwX

wPxwXwPxwXP)(

)())( with s' ofset ())((

Random Variables

• Example:– If two balanced die are rolled, then an outcome is a pair w=(i,j), where i

and j are integers between 1 and 6

– If X(w)=i+j, then this represents the random variable that computes the sum of the dice

– P(X(w)=7) =

Discrete Random Variable

• A random variable, X, is said to be discrete if its possible values may be arranged in a sequence, {x1, x2,…}

• E.g., X is the outcome of a roll of a die

• E.g., If X has non-negative integer values

Probability Function

• The probability function of a discrete random variable X(w) is

xwX

wPxwXPxf)(

)())(()(

Example

• If two balanced die are rolled, then an outcome is a pair w=(i,j), where i and j are integers between 1 and 6

• If X(w)=i+j, then this represents the random variable that computes the sum of the dice

• f(x)=

Example (2.1c)

• A coin is tossed until a heads appears

• The possible outcomes are Ω=

• Let X be the random variable denoting the trial for which the first heads is observed

• f(x)=

Properties of Probability Mass Functions

•

•

• Probability function demonstrates how the probability is distributed among the possible values of the random variable

• We refer to how it is distributed as the probability distribution

xxf allfor 0)(

1)(in

xfx

Joint Distributions

• Understanding the relationship among variables defined on the same sample space can be quite important

• If two random variables, X and Y, are defined on the same sample space, they are said to be jointly distributed

• To study their relationship we consider them together as a random vector (X,Y)

• The values of the random vector (x,y) have a joint probability (mass) function f(x,y)=P(X(w)=x,Y(w)=y)

Properties of the Joint Probability Mass Functions

•

•

•

yxyxf allfor , allfor 0),(

1),(in ),(

yxfyx

1),()(in

yxfxfy

Example (2.6)

• From a bowl containing 5 poker chips labeled 1-5, two are selected at random, one at a time, with replacement

• Let X denote the RV that describes the number on the first draw and Y denote the denote the RV that describes the number on the second draw

• Construct a joint probability table to display the distribution

Example (2.6)

• Find P(X=4,Y=5)

• Find P(X+Y=5)

• Find P(|X-Y|=2)

• Find P(X=1, Y=2 or 4)

Example (2.8)

• In a particular population, there may be a relationship between education and opinion on the death penalty

• The joint probabilities for the random variables education (X) and Opinion (Y) are given in the table below

Opinion Favor Oppose

Grade School 0.15 0.05 Education High School 0.20 0.10

College 0.25 0.25

Example (2.8)

• What is the probability that a randomly selected person has a high school education and opposes gun control?

Marginal Distributions

• Given the joint distribution of two random variables, the distribution of one of the variables alone is called the marginal distribution

• Marginal Distributions of X and Y:

–

–

),()()(in

yxfxXPxfy

X

),()()(in

yxfyYPyfx

Y

Example (2.6)

• Find P(Y=2)

• Find the marginal distribution of X

Example (2.8)

• Give the probability distribution for the variable Education

• Give the probability distribution for the variable Opinion

• Given that you know that the randomly selected individual has a college degree. What is the probability that they favor the death penalty

Independence

• Random variables X and Y are independent iff for every pair (x,y), f(x,y)=f(x)f(y)

• Same notion as before

Example (2.6)

• Are the variables X and Y independent

Example (2.8)

• Are the random variables Education and Opinion independent