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Chapter 16: Random Variables Lajja Majmundar & Vishmi Abeygunarathna Vocabulary ● Random Variable: A random variable assumes any of the several different values as a result of some random event. Random variables are denoted by a capital letter such as X. ● Discrete Random Variable: A random variable that can take one of a finite number of distinct outcomes is called a discrete random variable. ● Continuous Random Variable: A random variable that can take any numerical value within a range of values is called a continuous random variable. The range may be infinite or bounded at either or both ends. ● Probability Model: The probability model is a function that associates a probability P with each value of a discrete random variable X, denoted P(X=x), or with any interval of values of a continuous random variable. ● Expected Value: The expected value of a random variable is its theoretical long-run average value, the center of its model. Denoted μ or E(x), it is found (if random variable is discrete) by summing the products of variable values and probabilities. ● Variance: The variance of a random variable is the expected value of the squared deviation from the mean. ● Standard Deviation: The standard deviation of a random variable describes the spread in the model, and is the square root variance. Formulas ● Expected Value: ● Variance: E(X) = x1p1 + x2p2 + x3p3 + . . . + xnpn σ2 = Var(X) = Σ(x-μ)2P(x) ● Standard Deviation: σ = SD(X) = √Var(X) ● Adding/Subtracting Constants: E(X +/- c) = E(X) +/- c Var(X +/- c) = Var(X) ● Multiplying Variables: E(aX) = aE(X) Var(aX) = a2Var(X) ● Adding/Subtracting Variables: E(X +/- Y) = E(X) +/- E(Y) if X and Y are independent Var(X +/- Y) = Var(X) + Var(Y) Main Concepts ● A random variable is a value based on the outcome of a random event o outcomes can either be discrete or continuous based on how they are listed ● The collection of all possible values and the probabilities that they occur is called the probability model for the random variables ● The expected value can be calculated by multiplying each possible value by its probability and then finding the sum ● The variance is the expected value of the squared deviations ● Adding or subtracting a constant from data shifts the mean but does not change the variance or standard deviation o The same is true for random variables ● Multiplying each variable by a constant multiplies the mean by that constant and the variance by the square of that constant ● The expected value of the sum is the sum of the expected values o The variance of the sum of two independent random variables is the sum of their individual variances ● The mean of the sum of two random variables is the sum of the means ● The mean of the difference of two random variables is the difference of the means o If the random variables are independent, the variance of their sum of difference is always the sum of the variances ● When two independent continuous variables have Normal models, so does their sum or difference HW Problem #23 Random Variables. Given independent random variables with means and standard deviation as shown, find the mean and standard deviation of each of these variables. a.) 3X b.) Y+ 6 Mean SD c.) X + Y X 10 2 d.) X - Y e.) X1 + X2 Y 20 5 a.) μ = E(3X) = 3E(X) = 3(10) = 30 σ = SD(3X) = 3SD(x) = 3(2) = 6 b.) μ = E(Y + 6) = E(Y) + 6 = 20 + 6 = 26 σ = SD(Y + 6) = SD(Y) = 5 c.) μ = E(X + Y) = E(X) + E(Y) = 10 + 20 = 30 σ = SD(X + Y) = sqrt(Var(X) + Var(Y)) = sqrt(22 + 52) = 5.39 d.) μ = E(X - Y) = E(x) - E(Y) = 10 - 20 = -10 σ = SD(X - Y) = sqrt(Var(X) + Var(Y)) = sqrt(22 + 52) = 5.39 e.) μ= E(X1 + X2) = E(X) + E(X) = 10 + 10 = 20 σ = SD(X1 + X2) = sqrt(Var(X) +Var(X)) = sqrt(22 + 22) = 2.83 HW Problem #25 Random Variables. Given independent random variables with means and standard deviation as shown, find the mean and standard deviation of each of these variables. a.) 0.8Y b.) 2x - 100 Mean SD c.) X + 2Y X 120 12 d.) 3X - Y e.) Y1 + Y2 Y 300 16 a.) μ = E(.8Y) = .8E(Y) = .8(300) = 240 σ = SD(.8Y) = .8SD(Y) = .8(16) = 12.8 b.) μ = E(2X - 100) = 2E(X) - 100 = 140 σ = SD(2X - 100) = 2SD(X) = 2(12) = 24 c.) μ = E(X + 2Y) = E(X) + 2E(Y) = 120 + 2(300) = 720 σ = SD(X + 2Y) = sqrt(Var(X)+ 22Var(Y)) = sqrt(122 + 22(162) = 34.18 d.) μ = E(3X - Y) = 3E(X) - E(Y) = 3(120) - 300 = 60 σ = SD(3X - y) = sqrt(32Var(X)+ Var(Y)) = sqrt(32(122) + 162) = 39.40 e.) μ = E(Y1 + Y2) = sqrt(Var(Y) + Var(Y)) = sqrt(162 + 162) = 22.63