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# handout_4_4-1 - 4.4 Means and Variances of Random Variables...

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4.4 Means and Variances of Random Variables Mean of a discrete random variable Suppose that X is a discrete random variable whose distribution is Value of X x 1 x 2 x 3 x k Probability p 1 p 2 p 3 p k To find the mean of X, multiply each possible value by its probability, then add all the products. μ x = x 1 p 1 + x 2 p 2 + x 3 p 3 + …+ x k p k μ x =Σ x i p i the mean of a random variable is also frequently referred to as the expected value of a random variable: E(X) E(X) = μ x =Σ x i p i Example lottery- choose a 3-digit number (000-999) probability 1/1000 of winning \$500 X is a random variable- the amount your ticket pays probability distribution of X: Payoff X \$0 \$500 Probability 0.999 0.001 What is the average outcome from many tickets (long-run average payoff)? ●play lottery several times- call the mean of the actual amounts you win x ●long-run average outcome= mean of random variable = μ x

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Figure 4.13 If first digits in a set of data appear “at random,” all nine digits have the same probability. The probability distribution of the first digit X is: 1 st digit 1 2 3 4 5 6 7 8 9 prob 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 μ x = 1 × (1/9) + 2 × (1/9) + … μ x = 5 If first digits obey Benford’s law, the distribution of the first digit V is: 1 st digit 1 2 3 4 5 6 7 8 9 prob 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046 μ v = 1 × (0.301) + 2 × (0.176) + … μ v = 3.441 2
Statistical estimation and the law of large numbers The population distribution of a variable is the distribution of its values for all members of the population. The population distribution is also the probability distribution (sampling distribution ) of the variable when we choose one individual at random from the population. sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population EX consider the heights of young women suppose the population = 1000/ then suppose take SRS of size n = 1 population distribution→ 1000 possible values sampling distribution→ 1000 possible samples of size 1 from population of 1000 sampling distribution→ 1000 possible values The distribution of heights of young women (18-24) is N(64.5, 2.5). Select a young woman at random and measure her height (random variable X). In repeated sampling X will have the same N(64.5, 2.5) distribution. population distribution= sampling distribution when we choose one individual at random from population 3

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We want to estimate the mean height (μ) of all American women (18-24). This μ = μ
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handout_4_4-1 - 4.4 Means and Variances of Random Variables...

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