Course Notes - Department of Mathematics and Statistics

X = x i Pr(X = x i )0 0.101 0.252 0.403 0.204 0.05Meanµ X = 0(0.10) + 1(0.25) + 2(0.40) + 3(0.20) + 4(0.05) = 1.85Varianceσ 2 X = (0 − 1.85)2 × 0.10 + (1 − 1.85) 2 × 0.25 + (2 − 1.85) 2 ×0.40 + (3 − 1.85) 2 × 0.20 + (4 − 1.85) 2 × 0.05 = 1.0275Standard Deviationσ X =√σ 2 X = √ 1.0275 = 1.0137Combining Random Variables• Often we are interested in the mean and the variance of a rescaledrandom variable, or in the mean and variance of sums (or differences)of random variables.• We will look at some properties of random variables (both discreteand continuous)Modifying Random Variables• If X is an independent random variable and a and b are constants.• Consider a new random variable Y, where:Y = a + bX• The mean of Y can be calculated by:µ Y = a + bµ X• The variance of Y can be calculated by:σY 2 = b2 σX252