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. Ifisthe length in cm andisthe length in inches, then

If the expected result ofmeasuring the length in inches isandthe expected result of measuring the length in cm isthen

If the variance of all thelengths measured in inches isthenthe variance of all the lengths measured in cm is

The variableinthe example above is a transformation of the variable

In general if a variableistransformed to give a new variableusingthe rulewhereis 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)

sinceandareindependent.

Instead we write

and

Ifandaredrawn from the same populatiion thensoandsoandthis is not equal to (1).

In general, we take a linearcombination of quite different random variablesandTherules for a linear combination of random variables are

and