## The Different Treatments of a Transformation of a Random Variable and a Combination of Random Variables

It is often desirable torecalibrate the unites in which a quantity is measured – forexample from inches to cm. There are about 2.58 cm in each inch, so alength in cm with be 2.58 times the same length in inches. If isthe length in cm and isthe length in inches, then If the expected result ofmeasuring the length in inches is andthe expected result of measuring the length in cm is then If the variance of all thelengths measured in inches is thenthe variance of all the lengths measured in cm is The variable inthe example above is a transformation of the variable In general if a variable istransformed to give a new variable usingthe rule where is a constant, then and A diferent treatment isrequired for a sum of INDEPENDENT random variables. If two randomvariables X-1 and X-2 are combined to give a third randomvariable, thenwe cannot write and (1)

since and areindependent. and If and aredrawn from the same populatiion then so and so andthis is not equal to (1).
In general, we take a linearcombination of quite different random variables and  Therules for a linear combination of random variables are and  