Constant times a random variable question

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Discussion Overview

The discussion revolves around the effects of multiplying a random variable by a constant on its distribution. Participants explore whether the new random variable retains the same distribution or belongs to the same family, considering different cases for the constant.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant states that if a random variable X follows a distribution FX, then multiplying by a constant a results in a new random variable aX, which does not follow the same distribution but belongs to the same family, provided a > 0.
  • Another participant notes that if a < 0, the new random variable FY may not belong to the same family as FX.
  • A participant mentions that if X is normally distributed, then cX is also normally distributed for any nonzero real number c, and similarly for X + d, for any real number d.
  • One participant suggests using moment generating functions (mgf) to determine if the distribution remains unchanged, indicating that if the mgf structure is the same, the distribution does not change.
  • Another participant elaborates on using mgf to analyze the effects of scaling and translating a normal distribution, stating that scaling changes the variance and translating adds to the mean.

Areas of Agreement / Disagreement

Participants express differing views on the implications of multiplying a random variable by a constant, particularly regarding the conditions under which the new variable retains the same distribution or family. No consensus is reached on the broader implications for distributions beyond the normal case.

Contextual Notes

Participants highlight the importance of the sign of the constant a and its implications for the distribution. The discussion also touches on the use of moment generating functions as a tool for analysis, but specific mathematical steps and assumptions remain unresolved.

jimmy1
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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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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.
 
Last edited:
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
 
jimmy1 said:
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.
 
mheena said:
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

Like I said to the previous poster, use moment generating functions. Let U = cX + d. Use E[e^(tU)] be the mgf of U and you will find that this will give the mgf of a normal distribution with different parameters which will tell you what scaling and translating a normal does to its parameters. (Translating adds to the mean, scaling changes the variance by multiplying it by c^2).
 

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