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Jan 16

Scaling a distribution

Scaling a distribution is as important as centering or demeaning considered here. The question we want to find an answer for is this: What can you do to a random variable X to obtain another random variable, say, Y, whose variance is one? Like in case of centering, geometric considerations can be used but I want to follow the algebraic approach, which is more powerful.

Hint: in case of centering, we subtract the mean, Y=X-EX. For the problem at hand the suggestion is to use scaling: Y=aX, where a is a number to be determined.

Using the fact that variance is homogeneous of degree 2, we have

Var(Y)=Var(aX)=a^2Var(X).

We want Var(Y) to be 1, so solving for a gives a=1/\sqrt{Var(X)}=1/\sigma(X). Thus, division by the standard deviation answers our question: the variable Y=X/\sigma(X) has variance and standard deviation equal to 1.

Note. Always use the notation for standard deviation \sigma with its argument X.

One Response for "Scaling a distribution"

  1. […] we know that that can be achieved by centering and scaling. Combining these two transformations, we obtain the definition of the z […]

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