# Constant times a random variable question

A random variable X follows a certain distribution. Now say I multiply the random variable X by a constant a. Does the new random variable aX follow the same distribution as X?

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EnumaElish
Homework Helper
Well, let's take it through the basics. If X ~ FX, then FX(x) = Prob{X < x} for x < X < $\bar x$. Let Y = aX for ax < Y < a$\bar x$; then FY(y) = Prob{Y < y} = Prob{aX < y} = Prob{X < y/a} = FX(y/a). Therefore FY(y) = FX(y/a) for ax < Y < a$\bar x$.

Is FY the identical distribution as FX? No. Does it belong to the same family as FX? Yes, up to a scaling factor.

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EnumaElish
Homework Helper
Note: in the previous post I assumed a > 0. If a < 0, then FY may not even belong to the same family as FX.

The case of a = 0 is trivial, but you should bear that in mind, too.

we know that if X is normally distributed, then so cX for any nonzero real number c.
also X + d is normally distributed, for any real number d.
can anyone please show me the proof?thanks

chiro
A random variable X follows a certain distribution. Now say I multiply the random variable X by a constant a. Does the new random variable aX follow the same distribution as X?
If you want a systematic way to figure this out, use moment generating functions and if the structure of the mgf is the same as the unscaled distributions mgf, then you know that the distribution doesn't change and you can see how the parameters of the distribution have changed.

chiro