Understanding Uniformly Distributed Random Variables

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Discussion Overview

The discussion centers on the properties of uniformly distributed random variables, specifically examining the transformation of a uniformly distributed random variable P ~ U(1,2) when multiplied by a constant x. Participants explore whether the resulting variable xP retains the same uniform distribution or if its range changes.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions if xP ~ U(1,2) remains valid when P is uniformly distributed over (1,2), suggesting uncertainty about the impact of multiplying by a constant.
  • Another participant provides a probability calculation for P, stating that Prob(P < p) = p - 1, and begins to derive the probability for xP under the assumption that x > 0.
  • A similar point is reiterated by another participant, who also explores the probability transformation and concludes that xP might still follow a uniform distribution, though they acknowledge their assumptions.
  • In contrast, a different participant asserts that if U(1,2) is the uniform distribution on (1,2), then xP would actually be distributed according to U(x,2x), indicating a change in the range.

Areas of Agreement / Disagreement

Participants express differing views on whether xP retains the uniform distribution U(1,2) or transforms to U(x,2x). There is no consensus on the outcome of this transformation.

Contextual Notes

Participants rely on specific assumptions about the constant x (e.g., x > 0) and the nature of the uniform distribution, which may affect their conclusions. The discussion does not resolve the implications of these assumptions.

roadworx
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If I have random variable, P ~ U(1,2), am I correct in thinking that xP ~ U(1,2) also ? (where x is some constant), or does the range change?

Thanks.
 
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Since P is U(1, 2), Prob(P < p) = p - 1. Note Prob(P < 1) = 1 - 1 = 0 and Prob(P < 2) = 2 - 1 = 1.

Assume x > 0, then Prob(xP < p) = Prob(P < p/x) = ...

Does this help?
 
Last edited:
EnumaElish said:
Since P is U(1, 2), Prob(P < p) = p - 1. Note Prob(P < 1) = 1 - 1 = 0 and Prob(P < 2) = 2 - 1 = 1.

Assume x > 0, then Prob(xP < p) = Prob(P < p/x) = ...

Does this help?

So Prob(P<p/x) = p/x -1 ?

when p=x Prob = 0, and 2p = x, Prob = 1

So xP still follows a Uniform Distribution ~ U(1, 2).

It looks as though I've assumed this though.
 
if U(1,2) is the uniform distribution on (1,2), then the random variable xP would be distributed according to U(x,2x).

Torquil
 

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