handout_4_4 - 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 Figure 4.13
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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 We want to estimate the mean height (μ) of all American women (18-24). This μ = μ x of the random variable X obtained by choosing a young woman at random and measuring her height. To estimate μ, we choose an SRS of young women and use the sample mean x to estimate the unknown population mean μ. μ = parameter x = statistic We don’t expect x to be exactly equal to μ, and we realize that if we choose another SRS, the luck of the draw will probably produce a different x . If x is rarely exactly right and varies from sample to sample, why is it nonetheless a reasonable estimate of the population mean μ? 1)
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This note was uploaded on 02/23/2011 for the course STAT 2700 taught by Professor Bill during the Spring '11 term at Adelphi.

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

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