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7.2 Day 1: Mean & Variance of Random Variables

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?.

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7.2 Day 1: Mean & Variance of Random Variables

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  1. 7.2 Day 1: Mean & Variance of Random Variables Law of Large Numbers

  2. 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.

  3. 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.

  4. Since there are 1000 possible 3 digit numbers, your probability of winning is 1/1000.

  5. The probability distribution of X (the amount your ticket pays you) In the long run, you would only receive $500 once in every 1,000 tickets and $0 in the remaining 999 of the tickets Payoff X: $0 $500 Probability: 0.999 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.

  6. So what is the mean? We will say that μx= $0.50. • The long-run average payoff or mean for this random variable X is fifty cents. • This is also known as the Expected Value.

  7. 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: p1p2p3 … pk • To find the mean of X, multiply each possible vlaue by its probability, then add all the products • μx = x1p1 + x2p2 + … + xkpk • = Σxipi We will use μx to signify that this is the mean of a random variable and not of a data set.

  8. Ex 2: Benford’s LawCalculating the expected first digit The expected value is μx = 5. • What is the expected value of the first digit if each digit is equally likely? 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

  9. What is the expected value if the data obeys Benford’s Law? μx = 1(.301) + 2(.176) + 3(.125) + 4(.097) + 5(.079) + 6(.067) + 7(.058) + 8(.051) + 9(.046) = 3.441 The expected value is μx = 3.441.

  10. Probability Histogram for equally likely outcomes 1 to 9 In this uniform distribution, the mean 5 is located at the center.

  11. Probability Histogram for Benford’s Law The mean is 3.441 in this right skewed distribution.

  12. 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.

  13. Variance of a Discrete Random Variable • Suppose that X is a discrete random variable whose distribution is Value of X: x1 x2 x3 … xk Probability: p1p2p3 … 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. We will use σx2 to signify the variance and σx for the standard deviation.

  14. 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:

  15. Find the mean and variance. μx = 1.1 σx2 = 0.890 The standard deviation is σx = 0.943

  16. 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.

  17. Ex 4: Heights of Young Women(Law of Large Numbers) The average height of young women is 64.5 in.

  18. 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.

  19. 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.

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