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7.2 Day 1: Mean & Variance of Random Variables Law of Large Numbers The Mean of a Random Variable The mean x of a set of observations is their ordinary average, but how do you find the mean of a discrete random variable whose outcomes are not equally likely? The Mean of a Random Variable is known as its expected value. Ex 1: The Tri-State Pick 3 In the Tri-State Pick 3 game that New Hampshire shares with Maine and Vermont, you choose a 3-digit number and the state chooses a 3-digit winning number at random and pays you $500 if your number is chosen. Since there are 1000 possible 3 digit numbers, your probability of winning is 1/1000. The probability distribution of X In the long run, you (the amount your ticket payswould you) only receive Payoff X: Probability: $0 0.999 $500 once in every 1,000 tickets and $0 in the remaining 999 of the tickets $500 0.001 The ordinary average of the two possible outcomes is $250, but that makes no sense as the average because $0 is far more likely than $500. So what is the mean? We will say that μx = $0.50. 1 999 $500 $0 $0.50 1000 1000 The long-run average payoff or mean for this random variable X is fifty cents. This is also known as the Expected Value. Mean of a Discrete Random Variable Suppose that X is a discrete random variable whose distribution is Value of X: x1 x2 x3 … xk Probability: p1 p2 pWe … usepμk to 3 will x signify that this is To find the mean of X, multiply each possible the mean of a random variable vlaue by its probability, then add all the products and not of a data μx = x1p1 + x2p2 + … + xkpkset. = Σxipi Ex 2: Benford’s Law Calculating the expected first digit What is the expected of the first digit if The value expected each digit is equally likely? value is μ = 5. x First Digit X 1 2 3 4 5 6 7 8 9 Probability 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 μx = 1(1/9) + 2(1/9) + 3(1/9) + 4(1/9) + 5(1/9) + 6(1/9) + 7(1/9) + 8(1/9) + 9(1/9) =5 What is the expected value if the data obeys Benford’s Law? First Digit X Probability 1 .301 2 3 4 5 6 7 The expected .176 .125 .097 .079 .067 .058 value is μx = 3.441. 8 .051 μx = 1(.301) + 2(.176) + 3(.125) + 4(.097) + 5(.079) + 6(.067) + 7(.058) + 8(.051) + 9(.046) = 3.441 9 .046 Probability Histogram for equally likely outcomes 1 to 9 In this uniform distribution, the mean 5 is located at the center. Probability Histogram for Benford’s Law The mean is 3.441 in this right skewed distribution. Recall… Computing a measure of spread is an important part of describing a distribution (SOCS) The variance and the standard deviation are the measures of spread that accompany the choice of the mean to measure center. Variance of a Discrete Random Variable Suppose that X is a discrete random variable We will use σx2 to whose distribution is signify the variance Value of X: x1 x2 andxσ3x for… the xk Probability: p1 standard p2 p3 deviation. … pk And that the mean μ is the mean of X. The variance of X is σx2 = (x1 – μx)2p1 + (x2 – μx)2p2 + … + (xk – μx)2pk The standard deviation σx of X is the square root of the variance. Ex 3: Linda Sells Cars Linda is a sales associate at a large auto dealership. She motivates herself by using probability estimates of her sales. For a sunny Saturday in April, she estimates her car sales as follows: Cars Sold: 0 1 2 3 Probability: 0.3 0.4 0.2 0.1 Find the mean and variance. xi pi xipi (xi – μx)2pi 0 0.3 0.0 (0 – 1.1)2(0.3) = 0.363 The 0.004 standard deviation is σx = 0.943 1 0.4 0.4 (1 – 2 0.2 0.4 (2 – 1.1)2(0.2) = 0.162 3 0.1 0.3 (3 – 1.1)2(0.1) = 0.361 μx = 1.1 1.1)2(0.4) = σx2 = 0.890 The Law of Large Numbers 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 mean x of the observed values eventually approaches the mean μ of the population as closely as you specified and then stays that close. Ex 4: Heights of Young Women (Law of Large Numbers) The average height of young women is 64.5 in. The Law of Small Numbers The law of small numbers does not exist, although psychologists have found that most people believe in the law of small numbers. Most people believe that in the short run, general rules of probability with be consistent. This is a misconception because the general rules of probability only exist over the long run. In the short run, events can only be characterized as random. How large is a large number? The law of large numbers does not state how many trials are necessary to obtain a mean outcome that is close to μ. The number of trials depends on the variability of the random outcomes. The more variable the outcomes, the more trials that are needed to ensure that the mean outcome x is close the distribution mean μ.