The inverse of uniform random variable

In summary, chiro said that you can calculate the expected value of Y using E[X] and that you can use transformation theorems to calculate it. He said that you need to know the distribution of X before you can calculate E[1/X]. He also said that he will try to make the explanation more clear in a new thread.
  • #1
giglamesh
14
0
Hi all
I'm looking for solving this problem to find the closed form solution if it is possible:

[itex]Y=\frac{1}{X}[/itex]

Where X is uniform random variable > 0
I know the expected value for X which is [itex]\overline{X}[/itex]

is there a method to find the expected value of Y which is [itex]\overline{Y}[/itex] in term of [itex]\overline{X}[/itex] as closed form solution?

I know how to calculate it easily using numerical solution, but I need it for modeling problem and I need the analytical solution.

Thanks
 
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  • #2
giglamesh said:
Hi all
I'm looking for solving this problem to find the closed form solution if it is possible:

[itex]Y=\frac{1}{X}[/itex]

Where X is uniform random variable > 0
I know the expected value for X which is [itex]\overline{X}[/itex]

is there a method to find the expected value of Y which is [itex]\overline{Y}[/itex] in term of [itex]\overline{X}[/itex] as closed form solution?

I know how to calculate it easily using numerical solution, but I need it for modeling problem and I need the analytical solution.

Thanks

Hey giglamesh and welcome to the forums.

What is your statistics and probability background like?

A standard intro year course in university will give you the tools to solve this problem. Do you know about transformation theorems in statistics?

http://www.ncur20.ws/presentations/2/216/presentation.pdf
 
  • #3
hello chiro
Thanks for replying

I have a background with Random Variables and stochastic processes
I've read about MLE once but never use it in my applications, I remember that it is used to estimate the random variable from sample data vectors.

which is not what I'm looking for.

Maybe I didn't explain my problem well:

X is random variable I know only it's expected variable
Y=1/X is a random variable I need to know it's expected value using only E[X]

I'll check the transformation methods you mentioned, I know there are Laplace and Z-transform, I've used Z-transform but it didn't give the required result.

I'll try to search more.
Thanks
 
  • #4
Aah, that changes it. Given a continuous random variable X with pdf [itex]f_X[/itex] and a function g, we can always calculate

[tex]E[g(X)]=\int_{-\infty}^{+\infty}{g(x)f_X(x)dx}[/tex]

So in your case, you need to calculate

[tex]E[1/X]=\int_{-\infty}^{+\infty}{\frac{f_X(x)}{x}dx}[/tex]

So if X is uniform(1,2) for example, then

[tex]E[1/X]=\int_1^2{\frac{1}{x}dx}=\log{2}[/tex]
 
  • #5
hi micromass
Actually X is discrete I need to say: X is not uniform but Y=1/X is constructed as a uniform distribution from X, that means gives that X=3 then Y=1/3
P(y)=E[1/X]
I know only X then I need to get E[1/X] using only E[X] which is known but the distribution of X is not known.

Thanks for replying
 
  • #6
I think what chiro said makes sense for me right now

X 0 1 2 3 ...H
P(Y|X=i)=1/i 1 0.5 1/3 ...1/H

From the second line I'll try to estimate the PMF of Y using MLE, I'll try it
 
  • #7
I just closed this thread, I will open new one and try to make it more clear.
 

1. What is the inverse of a uniform random variable?

The inverse of a uniform random variable is a function that maps the range of the uniform distribution to a new range, typically between 0 and 1. It essentially "reverses" the process of generating a random number from a uniform distribution.

2. How is the inverse of a uniform random variable calculated?

The inverse of a uniform random variable is calculated by taking the cumulative distribution function (CDF) of the uniform distribution and solving for the input variable. This can be done using algebraic manipulation or by using a lookup table or algorithm.

3. What is the purpose of the inverse of a uniform random variable?

The inverse of a uniform random variable is commonly used in simulations and data analysis to generate random numbers with a specific distribution. It allows for the creation of random variables that follow a uniform distribution, which is useful for modeling real-world scenarios.

4. Can the inverse of a uniform random variable be used for any type of distribution?

No, the inverse of a uniform random variable can only be used for uniform distributions. For other distributions, such as normal or exponential, different methods must be used to generate random numbers with those distributions.

5. What is the relationship between the inverse of a uniform random variable and the probability density function (PDF)?

The inverse of a uniform random variable is related to the PDF of the uniform distribution through the derivative of the CDF. The PDF represents the probability of a random variable falling within a certain range, while the inverse function allows for the calculation of the input variable given a certain probability.

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