Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Chap 7.2 - West Ada
Document related concepts
Transcript
Letβs define the random variable π = πππ‘ ππππ ππππ π π πππππ $1 πππ‘ ππ πππ. 1. Is this example a discrete or continuous random variable? 2. What are the possible values of π? 3. Create a probability distribution. 4. What is the playerβs average gain? Chap 7.2 Means and Variance from Random Variables Mean of a Random Variable ο΅ An average of the possible values of βπβ, but with an essential change to take into account the fact that not all outcomes are equally likely ο΅ Also called the βExpected Valueβ Mean of a Discrete Random Variable (Expected Value) ο΅ To find the mean of βπβ, multiply each possible value by its probability, then add all the products ο΅ ππ = π₯1 π1 + π₯2 π2 + β― + π₯π ππ = ο΅ Back to the warm upβ¦.. ο΅ π₯π ππ What is the playerβs average gain? ο΅ Example 7.5 Most states and Canadian provinces have governmentsponsored lotteries. Here is a simple lottery wager, from the Tri-State Pick 3 game that New Hampshire shares with Maine and Vermont. You choose a three-digit number; the state chooses a three-digit winning number at random and pays you $500 if your number is chosen. ο΅ How many three-digit combinations are there to chose from? ο΅ What is the probability of winning? ο΅ Let X be the amount your ticket pays you, draw a probability distribution. ο΅ What is your average pay-off from many tickets? Law of Large Numbers ο΅ Draw Independent observations at random from any population with finite mean µ. Decide how accurately you would like to estimate µ. As the number of observations drawn increases, the same mean π₯ of the observed values eventually approaches the mean µ of the population as closely as you specified and then stays that close. ο΅ Nutshell: (in the long run) ο΅ ο΅ Proportion of outcomes get closer to the probability of that value ο΅ Average outcome gets closer to the distribution mean βLaw of small numbersβ is a mythβ¦. ο΅ Heads and tailsβ¦. Rules for Means 1. If π is a random variable and π and π are fixed numbers then ο΅ 2. ππ+ππ = π + πππ If π₯ and π¦ are random variables, then ο΅ ππ₯+π¦ = ππ₯ + ππ¦ Variance of a Random Variable ο΅ Variance of a Discrete Random Variable ο΅ Example 7.7 pg. 485 ο΅ The Standard Deviation (ππ₯ ) of π₯ is the square root of the variance ο΅ ππ₯2 = ( π₯1 β ππ )2 π1 + ( π₯2 β ππ )2 π2 + β― + π₯π β ππ 2 = ( π₯π β ππ )2 ππ You can not add the standard deviations, but you can add the variances and then evaluate for the standard deviations. Rules for Variance 1. If π is a random variable and π and π are fixed numbers then ο΅ 2. 2 ππ+ππ = π2 ππ2 If π and π are independent random variable then 2 ο΅ ππ+π = ππ2 + ππ2 2 ο΅ ππβπ = ππ2 + ππ2 and Assignment: pg.486 23-25, 32, 34, 38-40 Random trivia: Some people have all the luckβ¦ In June 2005, Donna Goeppert won a million dollar jackpot playing the Pennsylvania Lottery. The odds of winning a lottery like the one in Pennsylvania are 1.44 million to 1. But what is extraordinary is that Ms. Goeppert had previously won $1 million playing the lottery earlier in the year. The odds of winning twice vary, depending on how many tickets are scratched. A university professor in Pennsylvania estimated that if you played 100 tickets, the odds of winning the lottery twice are about 419 million to 1.
