Stat 5101 Lecture Notes - School of Statistics

2.4. MOMENTS 43In contrast, for all practical purposes standard deviation is the preferredconcept, as evidenced by the fact that introductory statistics textbooks thatchoose to use only one of the two concepts invariably choose standard deviation.The reason has to do with units of measurement and measurement scales.Suppose we have a random variable X whose units of measurement are inches,for example, the height of a student in the class. What are the units of E(X),var(X), and sd(X), assuming these quantities exist?The units of an expectation are the same as the units of the random variable,so the units of E(X) are also inches. Now var(X) is also just an expectation,the expectation of the random variable (X − µ) 2 , so its units are the units of(X − µ) 2 , which are obviously inches squared (or square inches, if you prefer).Then obviously, the units of sd(X) are again inches. Thus X, E(X), and sd(X)are comparable quantities, all in the same units, whereas var(X) isnot. Youcan’t understand what var(X) tells you about X without taking the square root.It’s isn’t even in the right units of measurement.The theoretical emphasis of this course means that we will be primarilyinterested in variances rather than standard deviations, although we will beinterested in standard deviations too. You have to keep in mind which is which.StandardizationGiven a random variable X, there is always a linear transformation Z =a + bX, which can be thought of as a change of units of measurement as inExample 2.3.1, that makes the transformed variable Z have mean zero andstandard deviation one. This process is called standardization.Theorem 2.15. If X is a random variable having mean µ and standard deviationσ and σ>0, then the random variableZ = X − µ(2.17)σhas mean zero and standard deviation one.Conversely, if Z is a random variable having mean zero and standard deviationone, µ and σ are real numbers, and σ ≥ 0, then the random variablehas mean µ and standard deviation σ.X = µ + σZ (2.18)The proof is left as an exercise (Problem 2-17).Standardization (2.17) and its inverse (2.18) are useful in a variety of contexts.We will use them throughout the course.2.4.4 Mixed Moments and CovariancesWhen several random variables are involved in the discussion, there areseveral moments of each type, as we have already discussed. If we have two