# Thread: Uniform Distribution Proof

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

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

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

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

$\displaystyle = 1 - 1/v$

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!

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

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

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

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

$\displaystyle = 1 - 1/v$

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

CB

3. If U is a random-uniform from 0 to 1 - then $\displaystyle P(U \leq u) = u$. Similarly, the $\displaystyle P(U \geq u) = 1-P(U \leq u) = 1-u$.

4. 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.

5. cheers guys.... that helps a lot.

6. 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.

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