STAT 211 Topic 5

From Notes
Jump to navigation Jump to search
Lecture 11 Notes

« previous | Tuesday, March 1, 2011 | next »


Review

Topic 3: Discrete Random Variables

  • PMF, CDF
  • Expected value and variance

Topic 4: Continuous Random Variables

  • PDF, CDF
  • Epxected value and variance


Topic 5 Overview

Consider 2 or more random variables.

  • Joint PMF or PDF
  • Expected value and variance

Central Limit Theorem (CLT)


What is Joint Distribution?

Discrete

Given 2 Bernoulli discrete random variables and (not independent)

Joint Probability Mass Function
  1. Marginal PMF of X: (also satisfies 1 & 2)
  2. Marginal PMF of Y: (also satisfies 1 & 2)

Continuous

Let and be continuous random variables:

Note that whatever comes first in is the outer integral

Marginal Cases:

Example


Independence

if discrete
if continuous
You should be able to factor joint PDF into two functions: alone and alone

For example, if and are independent, .


Conditional Distributions

If and are continuous, the result of the above equation will be since is set to a certain constant number.

The way to get around it is by looking at the joint PDF and marginal PDF:

After finding symbolic conditional PDF, integrate over X and plug in Y


Example

Suppose and denote the air pressure in the front tires of a car (left and right) and their joint PDF is given by:

The tires are supposed to be filled to 26 psi.

  1. What is ?
    do double integral, set equal to 1, and solve for K
  2. What is the probability that both tires are underfilled?
    double-integrate from 20 to 26 on both integrals


Lecture 12

Lecture 12 Notes

Thursday, March 3, 2011

(continued from lecture 11)

Homework Example

Given 2 independent continuous rv's as Uniform distribution between 0 and 3, find

We don't know , but we do know the marginal PDFS:

By independence, the joint PDF is just the product of the two marginal PDFs:

Plot the function as a square with bottom left corner at origin and sides of length 3

Solve function inside probability () for and plot bounded area:

Find the area of the bounded area using either the graph or the integral above


Expected Value

For any arbitrary function

Easier way:

Variance

ONLY IF X AND Y ARE INDEPENDENT!


(lecture 12)

Random Sample

A collection of independent random variables with same distribution as population(recall from Topic 1) that can be used to estimate a parameter. The numbers we get to estimate parameters are called statistics

Central Limit Theorem

The original rv X may have any distribution.

If we let and , we know that:


Suppose we pick a random sample X1 through Xn from X:

As sample size gets larger, the average of the sample approaches a normal distribution:

Proof

Let X Be a uniform distribution

Take a 15 random samples of size 30. The expected value of the sample average should be the average of the population and have a normal distribution

Variance of the sample average should be the variance of the population average divided by the sample size:

Example 1

Flipping a fair coin ( for getting Heads or Tails)

Therefore by CLT, the average of 1000 flips will approach


Example 2

Suppose the weight of 50 yr-old males has a certain distribution with mean 150 lbs. and std. dev. 32 lbs. What is the approximate probability that the sample mean weight for a random sample of 64 is 160 lbs?