# Thread: Calculate inverse of CDF

1. ## Calculate inverse of CDF

Hello,
How can write 'Inverse CDF' with the below CDF?
Thanks

2. ## Re: Calculate inverse of CDF

you can't

this CDF is a many to one mapping and thus it's inverse, if it existed, would be a one to many mapping. A one to many mapping isn't a valid function.

3. ## Re: Calculate inverse of CDF

Thank you,but i have an answer but i don't know with which formula or trick is solved?

4. ## Re: Calculate inverse of CDF

But it isn't. $F(22) = 0$, $F^{-1}(0) = 64$. So in other words $F^{-1} \left ( F(22) \right ) = 64$, not 22.

-Dan

5. ## Re: Calculate inverse of CDF

Could you told me how calculated this??

6. ## Re: Calculate inverse of CDF

Originally Posted by life24
Could you told me how calculated this??
I did make a mistake. But now I have a different problem.

I picked a number. x = 22.

In your first post you posted a formula for F(x). This gives F(22) = 0.

In your second post you posted a formula for $F^{-1}(u)$. Using this I get $F^{-1}(0)$ is undefined. So there is no inverse element for x = 22.

If you restate your original definition of $F^{-1}(u)$ to include u = 0 then you run into my original objection: $F(22) = 0$ as before, but now $F^{-1}(0) = 64$. It should give back our original number x = 22.

-Dan

7. ## Re: Calculate inverse of CDF

Originally Posted by topsquark
I picked a number. x = 22. In your first post you posted a formula for F(x). This gives F(22) = 0. In your second post you posted a formula for $F^{-1}(u)$. Using this I get $F^{-1}(0)$ is undefined. So there is no inverse element for x = 22. If you restate your original definition of $F^{-1}(u)$ to include u = 0 then you run into my original objection: $F(22) = 0$ as before, but now $F^{-1}(0) = 64$. It should give back our original number x = 22.
This particular given CDF is has the necessary properties: maps $(-\infty,\infty)$ monotonically increasing onto $[0,1]$ and is right-hand continuous.
However the pdf that determines it is finite (discrete): $\begin{array}{*{20}{c}}{64}&{128}&{512}\\\hline{0. 7}&{0.1}&{0.2}\end{array}$

I am greatly puzzled by this question. In years of teaching this material. I have seen the inverse of a CDF called for nor have I ever had the need for such. I would like to read the exact original question.

8. ## Re: Calculate inverse of CDF

In INVERSE TRANSFORMATION section for generate random variable.

9. ## Re: Calculate inverse of CDF

Originally Posted by life24
In INVERSE TRANSFORMATION section for generate random variable.
I guess you are in some sort of computer applications course?
Look at this page. As you can see there is no standard definition for inverse transform. Having taught probability/math-stat for many years, it is strange not to have ever seen the concept. If it is unique to your course you do well to find a computer science help site.

10. ## Re: Calculate inverse of CDF

Originally Posted by Plato
I guess you are in some sort of computer applications course?
Look at this page. As you can see there is no standard definition for inverse transform. Having taught probability/math-stat for many years, it is strange not to have ever seen the concept. If it is unique to your course you do well to find a computer science help site.
The idea of the inverse CDF comes up in hypothesis testing. You are generally given some max tail significance and a typical problem is to adjust the number of data samples until the sample variance is low enough to meet this. This requires finding the z-score given a probability which involves the inverse CDF.

For a discrete distribution like this you wouldn't really consider the actual values on the real line other than to define bins. These bins then become the discrete support for the distribution. There is a 1 to 1 map from a bin to a probability value.

11. ## Re: Calculate inverse of CDF

I see a fundamental problem with this question...

Does the area under the CDF equal 1???

12. ## Re: Calculate inverse of CDF

Originally Posted by Prove It
I see a fundamental problem with this question...

Does the area under the CDF equal 1???
it's the area under a PDF that must equal 1. The CDF doesn't have this restriction.

13. ## Re: Calculate inverse of CDF

[QUOTE=Prove It;910864]I see a fundamental problem with this question...
As romsek said that is a necessary property of a probability density function, PDF.
A CDF is a monotonically increasing, right-continuous function that maps onto $(-\infty,\infty)\to [0,1]$.

BTW. Thank you romsek for the clarification on the inverse transformation. However, I did check what I think of as two standard applied statistics texts. But neither of those discussed anything about that topic. What is a good reference for this topic?

14. ## Re: Calculate inverse of CDF

[QUOTE=Plato;910869]
Originally Posted by Prove It
BTW. Thank you romsek for the clarification on the inverse transformation. However, I did check what I think of as two standard applied statistics texts. But neither of those discussed anything about that topic. What is a good reference for this topic?
Papoulis has a section on hypothesis testing.

I think what background I have in it came from Poor's "An Introduction to Signal Detection and Estimation" which is much more focused on signal processing than on statistics in general.