Distribution of Fractional Polynomial of Random Variables

In summary: X_{1},X_{2},...,X_{n}$ $\ln |X_{1}|,\ln |X_{2}|,...,\ln |X_{n}|$ e^{-x} $P$ $e^{-x}$ $\mathcal{L}^{-1}$ $\frac{x^{n-1}}{(n-1)!}$
  • #1
liuch37
1
0
Hi all,

I would like to find the distribution (CDF or PDF) of a random variable Y, which is written as

Y=X_1*X_2*...X_N/(X_1+X_2+...X_N)^N.

X_1, X_2,...X_N are N i.i.d. random variables and we know they have the same PDF f_X(x).

I know this can be solved by change of variables technique and use multi-dimensional integration. But is there a easier way to deal with it?

Thanks.
 
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  • #2
liuch37 said:
Hi all,

I would like to find the distribution (CDF or PDF) of a random variable Y, which is written as

Y=X_1*X_2*...X_N/(X_1+X_2+...X_N)^N.

X_1, X_2,...X_N are N i.i.d. random variables and we know they have the same PDF f_X(x).

I know this can be solved by change of variables technique and use multi-dimensional integration. But is there a easier way to deal with it?

Thanks.

There is a 'milestone' that allows You to solve the problem : if $X_{1},X_{2},...,X_{n}$ are independent r.v. with p.d.f. $f_{1}(x),f_{2}(x),...,f_{n}(x)$, then the r.v. $X= X_{1} + X_{2} + ...+ X_{n}$ has p.d.f...

$$f(x) = f_{1} (x) * f_{2} (x) * ... *f_{n} (x)\ (1)$$

... where '*' means convolution. Now let's introduce for any r.v. $X_{i}$ the auxiliary r.v. $\Lambda_{i} = \ln |X_{i}|$ with p.d.f. $\lambda_{i} (x)$. Because is...

$$\ln |Y| = \ln |X_{1}| + \ln |X_{2}| + ... + \ln |X_{n}| - n\ \ln |X_{1} + X_{2} + ... + X_{n}|\ (2)$$

... You can apply (1). The only criticity is the fact that the r.v. $\sum_{i} \ln |X_{i}|$ and $n \ln |\sum_{i} X_{i}|$ are not independent.

An example: if the $X_{i}$ are all uniformely distributed in [0,1], what is the p.d.f. of $ Y = \sum_{i} \ln X_{i}$?... If X is uniformely distributed in [0,1], then...

$$P \{ \ln X < - x \} = \int_{e^{- x}}^{1} d\ \xi = 1 - e^{- x}\ (3)$$

Deriving (3) we find that the r.v. $\ln X$ has p.d.f. $e^{-x}$, which has Laplace Transform...

$$\mathcal{L} \{e^{-x} \} = \frac{1}{1 + s}\ (4)$$

Now we can find that the p.d.f. of Y ...

$$\lambda (x) = \mathcal {L}^{-1} \{ \frac{1}{(1+s)^{n}}\} = \frac{x^{n-1}}{(n-1)!}\ e^{-x}\ (5)$$

Kind regards

$\chi$ $\sigma$
 

1. What is a fractional polynomial of a random variable?

A fractional polynomial is a type of mathematical function that is used to model the relationship between two variables. In this case, the random variable is the input, and the fractional polynomial is the output. It is commonly used in statistical analysis to describe the distribution of data and make predictions.

2. How is the distribution of a fractional polynomial of a random variable determined?

The distribution of a fractional polynomial of a random variable is determined by the coefficients and powers of the polynomial. These values can be estimated using statistical methods such as maximum likelihood estimation or least squares regression.

3. What are the advantages of using fractional polynomials for modeling random variables?

Fractional polynomials offer a flexible and adaptable approach to modeling random variables. They can capture a wide range of non-linear relationships, and their coefficients can be easily interpreted and compared. Additionally, they require fewer assumptions than other modeling techniques, making them more robust and versatile.

4. Are there any limitations to using fractional polynomials for random variables?

Like any modeling technique, fractional polynomials have some limitations. They may not be suitable for all types of data and may not always provide the best fit. In addition, they can be more computationally complex and require larger sample sizes compared to simpler models.

5. How can the results of a fractional polynomial analysis be interpreted?

The results of a fractional polynomial analysis can be interpreted by examining the coefficients and their corresponding powers. The coefficients represent the magnitude and direction of the relationship between the variables, while the powers indicate the shape of the relationship (e.g. linear, quadratic, etc.). Additionally, diagnostic plots and statistical tests can be used to evaluate the adequacy of the model and make inferences about the data.

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