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A. P. STATISTICS LESSON 7.2 MEANS AND VARIANCES OF RANDOM VARIABLES ESSENTIAL QUESTION: How are means and variances found for random variables? • To find the mean of random variables. • To find the variance of random variables. Means and Variances of Random Variables Probability is the mathematical language that describes the long-run regular behavior of random phenomena. The probability distribution of a random variable is an idealized relative frequency distribution. The Mean of a Random Variable The mean x of a set of observations is their ordinary average. The mean of a random variable X is also an average of the possible values of X, but with an essential change to take into account the fact that not all outcomes need to be equally likely. Example 7.5 The Tri-State Pick 3 Page 407 Most states have lotteries where you choose a three digit number. If a state charges a dollar to play and $500 if you win, what is your average payoff from many tickets? Payoff X: Probability: $0 0.999 $500 0.001 $500 _ 1__ + $0 _999_ = $0.50 1000 1000 Mean and Expected Value Just as probabilities are an idealized description of long-run proportions, the mean of a probability distribution describes the long-run average outcome. The common symbol μ, the Greek letter mu. You will often find the mean of a random variable X called expected value of X. 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 , p3 , ….. pk To find the mean of X, multiply each possible value by its probability, then add all products; μ = x1 p1 + x2 p2+ …….xk pk =∑xi pi Example 7.6 Benford’s Law* Page 408 - 409 * It is a striking fact that the first digits of number in legitimate records often follow a a distribution known as Benford’s Law page 345 The Variance of Random Variable The mean is a measure of the center of a distribution. The variance and the standard deviation are the measures of spread of that accompany the mean to measure center. We write the variance of a random variable X as σ2. Variance of a Discrete Random Variable Suppose that X is a discrete random variable whose distribution is Value of X: x1 Probability: p1 x2 p2 x3 p3 … … xk pk and that μ is the mean of X. The variance of X is σ2x = (x1 – μx )2p1 + ( x2 – μx)2p + …+ ( xk – μx)2pk = ∑ ( xi – μx )2 pi The standard deviation σx of X is the square root of the variance. Example 7.7 Selling Aircraft Parts Page 411 Gain Communications sells aircraft communications units to both the military and the civilian markets. Next year’s sales depend on market conditions that cannot be predicted exactly. Gain follows the modern practice of using probability estimates of sales. The military division estimates its sales as follows: Units sold Probability: 1000 0.1 3000 5000 0.3 0.4 10,000 0.2