# Uniform Distribution Proof

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• Dec 20th 2010, 01:17 PM
padawan
Uniform Distribution Proof
Hi all,

I have been going through a proof, but can't seem to make sense of a step, I am sure its trivial, but I can't seem to see it!....

Let $U$ be a uniform variable on $[0,1]$ and let $V= 1/U$, find the density function of $V$.

Find CDF first: $F_V (v) = P(V \le v)$

$= P(1/U \le v)$

$= P(U \ge 1/v)$

$= 1 - 1/v$

Now its the last step I cant seem to see; where does the 1 come from? I can see teh rest of the proof from that point on, but have problems seeing what happened in this step...

Many thanks for reading!
• Dec 20th 2010, 01:57 PM
CaptainBlack
Quote:

Originally Posted by padawan
Hi all,

I have been going through a proof, but can't seem to make sense of a step, I am sure its trivial, but I can't seem to see it!....

Let $U$ be a uniform variable on $[0,1]$ and let $V= 1/U$, find the density function of $V$.

Find CDF first: $F_V (v) = P(V \le v)$

$= P(1/U \le v)$

$= P(U \ge 1/v)$

$= 1 - 1/v$

Now its the last step I cant seem to see; where does the 1 come from? I can see teh rest of the proof from that point on, but have problems seeing what happened in this step...

Many thanks for reading!

Draw a picture $P(U\ge a)=1-a$ (That it is linear in $a$ is obvious, $P(U\ge 0)=1$ and $P(U \ge 1)=0$ are also obvious)

CB
• Dec 20th 2010, 04:38 PM
ANDS!
If U is a random-uniform from 0 to 1 - then $P(U \leq u) = u$. Similarly, the $P(U \geq u) = 1-P(U \leq u) = 1-u$.
• Dec 20th 2010, 06:43 PM
ark600
As an aside, the uniform distribution is much under-used.
When it comes to fitting data the least-squares method is almost always used without thinking.
But, in many circumstances the L-1 norm is far more accurate( admittedly, when it is out, then it is far out), but it is very neglected.
• Dec 24th 2010, 12:32 PM
padawan
cheers guys.... that helps a lot.
• Dec 25th 2010, 11:07 PM
CaptainBlack
Quote:

Originally Posted by ark600
As an aside, the uniform distribution is much under-used.

It is used all the time in fact. All pseudo random number generators attempt to produce uniformly distributed numbers on [0,1), and they are then transformed to other distributions as required. This would possibly make the uniform distribution the most frequent use of any probability model.

Quote:

When it comes to fitting data the least-squares method is almost always used without thinking.
But, in many circumstances the L-1 norm is far more accurate( admittedly, when it is out, then it is far out), but it is very neglected.
But computationally intensive, so until possibly the last 20-30 years impractical.

CB