Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
7.2 Means and Variances of Random Variables Objective: Calculate the mean and standard deviation of random variables Understand the law of large numbers The mean of a Random Variable Ex: You pick a three digit number in the lottery. If your number matches the states number, you win $500. What are your average winnings? Probability Distribution Outcome Probability Probabilities are an idealized description of long-run proportions, so the mean of a probability distribution describes the____________________________________. Mean of a Discrete Random Variable Suppose X is a discrete random variable whose distribution is Value of X Probabiltiy x₁ p₁ x₂ p₂ x₃ p₃ .... .... 𝑥𝑘 𝑝𝑘 To find the mean of X, multiply each possible value by its probability, then add all the products: 𝜇𝑥 = 𝑥1 𝑝1 + 𝑥2 𝑝2 + 𝑥3 𝑝3 + ⋯ . +𝑥𝑘 𝑝𝑘 Ex: The distribution of the count X of heads in four tosses of a balanced coin. Numbers of Heads Probability 0 1 2 3 4 0.0625 0.25 0.375 0.25 0.0625 The expected value is: 𝜇𝑥 = Construct a histogram of the distribution. What do you notice about the mean? Ex: What is the mean number of inhabitants in an American household? Inhabitants 1 Proportion 0.25 of households 2 0.32 3 0.17 4 0.15 5 0.07 6 0.03 7 0.01 Complete 7.17-7.19 (pg.389) 7.17 7.18 7.19 Statistical estimation and the law of large numbers Law of large numbers Draw independent observations at random from any population with finite mean (μ). Decide how accurately you would like to estimate the mean. As the number of observations drawn increases, the mean of the observed values eventually approaches the mean of the population as closely as you specified and then stays that close. Describe this in your own words? Reece’s example: http://www.rossmanchance.com/applets/Reeses3/ReesesPieces.html Ex: Use the average height of women to explain this. The mean is 64.5 in with a standard deviation of 2.5 in. Rules for Means Rule 1: If X is a random variable and a and b are fixed numbers, then 𝝁𝒂+𝒃𝒙 = Rule 2: If X and Y are (independent) random variables, then 𝝁𝒙+𝒚 = Ex: Military divisions Units Sold Probability 1000 0.1 3000 0.3 5000 0.4 10000 0.2 Civilian Division Units Sold Probability 300 0.4 500 0.5 750 0.1 Let x= # of military units sold y= # of civilian units sold What is the mean number of military units sold? Civilian units sold? If a profit of $2000 is made on each military unit sold and $3500 is made on each civilian unit, what is the total mean profit for units sold? Variance of a Discrete Random Variable The variance of X is: Value of X Probabiltiy x₁ p₁ x₂ p₂ x₃ p₃ .... .... 𝑥𝑘 𝑝𝑘 𝜎𝑥2 = (𝑥1 − 𝜇𝑥 )2 𝑝1 + (𝑥2 − 𝜇2 )2 𝑝2 +(𝑥3 − 𝜇3 )2 𝑝3 + ⋯ + (𝑥𝑘 − 𝜇𝑘 )2 𝑝𝑘 So the standard deviation is: √𝜎2𝑥 Steps in the calculator: L₁ L₂ L₃ x p (𝐿1 − 𝜇𝑥 )2 𝐿2 Then sum(L₃) Find the standard deviation of the military units sold? Find the standard deviation of the civilian units sold? Rules for Variance Rule 1: 𝝈𝟐𝒂+𝒃𝒙 = Rule 2: 𝝈𝟐𝒙+𝒚 = ***___________________________________________________________________*** Ex: The payoff X of a $1 ticket in the Tri-State pick 3 game is $500 with probability 1/1000 and $0 the rest of the time. What is the variance of the total payoff if you buy $1 ticket on two different days? SAT scores SAT math score X SAT verbal score Y 𝜇𝑥 = 625 𝜎𝑥 = 90 𝜇𝑥 = 590 𝜎𝑥 = 100 What are the mean and standard deviation of the total score X + Y among students applying to this college? Golf scores Tom’s score X: 𝜇𝑥 = 110 𝜎𝑥 = 10 George’s score Y: 𝜇𝑥 = 100 𝜎𝑥 = 8 Their scores vary independently. What is the mean difference between their scores? What is the variance of the difference between their scores? So the standard deviation is? The class average on the last chapter test was 80 with a standard deviation of 4. What is the mean if I doubled everyone’s test score? What is the standard deviation? What if I added 5 bonus points to everyone’s score. What is the new mean and standard deviation? What if I doubled everyone’s score and added 5 points. What is the new mean and standard deviation?