A1 45730 29sep08

A1 45730 29sep08 - 45 730 Probability Decision Making...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
45 730 Probability & Decision Making (9/29/08) Topics for today’s class: • Central Limit Theorem • Some continuous distribution problems • Implications for Simulation
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Preliminaries to a last important result for the normal distribution: Consider n i.i.d. random variables X 1 , X 2 , ..., X n Definition: i.i.d. “independent and identically distributed” probability distribution is same for each X • mean and variance are the same for each X •a l l X ’s are independent of each other Issue: What can be said about the distribution of the sum of X 1 , X 2 , . .., X n ?
Background image of page 2
Some initial results: Consider the sum of n i.i.d. random variables X 1 + X 2 + . .. + X n • Result 1: Since identically distributed, mean of sum is sum of common mean E ( X 1 + X 2 + . .. + X n ) = ( μ + + . .. + ) = n • Result 2: Since independent, variance of sum is sum of (common) variances var( X 1 + X 2 + . .. + X n ) = ( σ 2 + 2 + . .. + 2 ) = n 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
And a very important result: The Central Limit Theorem (CLT) Suppose that X 1 , X 2 , . .., X n are n i.i.d. (independent and identically distributed) random variables Then as n grows large () 2 12 ... , n XX X N n n μ σ ++ + 2 1 ... , n X N nn ⎛⎞ + ⎜⎟ ⎝⎠ The sum (average) of i.i.d. random variables is approximately normally distributed
Background image of page 4
The CLT is one of the fundamental results in probability and statistics Why? • Provides rationale for why normal distribution is applicable for many uncertain situations • Allows use of normal distribution to approximate non-normal random variables (binomial, Poisson) • Serves as the foundation for statistical inference drawing conclusions about underlying probability model from observed data
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The CLT as a rationale for the normal distribution: Four examples: (in decreasing order of relevance) 1. Aggregate consumption of gasoline 2. Total sales by salespeople at a firm 3. Price of a single asset 4. Total return on a portfolio of many assets Common feature of these and many other examples: All involve sum of many (more or less) i.i.d. components normal distribution may be applicable due to CLT
Background image of page 6
The CLT as an approximation for non- normal random variables Recall: • Binomial random variable involves sum of n independent Bernoulli trials each with success probability p • Since sum of Poisson random variables is also a Poisson random variable, a particular Poisson random variable can be treated as the sum of lots of underlying Poisson processes
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Implication of the CLT for these situations: Since both binomial and Poisson random variables involve sums of many i.i.d. components, can use the normal distribution to approximate probabilities for those types of random variables A comment: With improvements in computing power, these results are no longer as important
Background image of page 8
Why is inference based on the CLT interesting/useful: • Will have situations where mean for probability model is not known, but observe sample of data from that model use CLT and estimates from observed data to form confidence intervals for mean Statistics!!!! (next mini) • Will have complex situations where issue of interest is mean outcome for some variable use CLT and outcomes from simulated data to form confidence intervals for mean outcome
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/05/2010 for the course BUS 45730 taught by Professor Ravi during the Fall '08 term at Carnegie Mellon.

Page1 / 64

A1 45730 29sep08 - 45 730 Probability Decision Making...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online