0
votes
1answer
15 views

Let $X_1$ and $X_2$ be independent $n(0,1)$ random variables. Find the pdf of $(X_1-X_2)^2/2$.

I understand that $(X_1-X_2)/\sqrt2)$ ~ $n(0,1)$ since it is a linear combination of $X_1 $ and $X_2$ and hence $(X_1-X_2)^2/2$ ~ $\chi^2_1$. I'm having trouble on how to prove/show this ...
1
vote
1answer
98 views

Relation between two distributions expressed in terms of their CDFs

Not great at stats, and having trouble wrapping my mind around this. Would love an explanation, not overly detailed, in plain english of what these transformations mean. The bias correction ...
1
vote
1answer
14 views

Will statistical analysis of transformed data hold for the original one?

I have a data with distribution like chisq-squared one. But ANOVA and t-test need the data to be normal distributed. So I want to do the Box-cox transformation to the data, but my concern is will the ...
0
votes
0answers
36 views

One double integral elated problem

The bit I am stuck is the limits in the double integral. I tried X from 0 to uy and Y from 0 to infinity, this is obviously incorrect. I just want to know the complete double integral in the order ...
0
votes
0answers
18 views

PCA - How to calculate the scores

I'm currently learning Principle component analysis and I have, so far calculated the Eigen values and vectors. Assume that I have the following: $$ E = \begin{pmatrix} 1 & 2\\ 3& 4 ...
2
votes
1answer
77 views

Variance stabilization for Poisson data

Intro Let $Z > 0$ be a random variable with the mean and variance defined as $\mathbb{E}\{ Z \}$ and $\operatorname{Var}\{ Z \}$, respectively. The variance stabilization transform (VST) $f(z)$ ...
1
vote
1answer
118 views

Joint density of two functions of random variable

This is online homework, and I'm not always clear on which chapter questions are from, so I might be completely off base. I have two random variables, $X_1$~UNI(5,10) and $X_2$~UNI(4,10), and then ...
0
votes
1answer
51 views

Take -log of a Beta distributed R.V.

X1.....Xn~Beta(a,1) Y = -log(X) Use the transformation formula to calculate the pdf of Y. What named distribution does it have? I am confused what method to use here. A beta does not converge to a ...
0
votes
0answers
13 views

stabilizing variance

Let $X_n$ be a Markov sequence such that $X_n=(1+X_{n-1})Y_n$ with $X_0=0$ and $Y_n$ - iid and independent of $X_n$'s. Suppose $\mathbb{E}[Y_1]=1$ and $\mathbb{E}[Y_1^2]=\varkappa$ with ...
0
votes
0answers
34 views

Derivation of F distribution

Prove that the PDF of Snecdor's F distribution, given by: $$F=\frac{U/n_1}{V/n_2}$$ Where $U=\chi^2(n_1)$ and $V=\chi^2(n_2)$, is given by: ...
0
votes
1answer
80 views

Question on transformations

Two efficiency experts take independent measurements Y1 and Y2 on the length of time workers take to complete a certain task. Each measurement is assumed to have the density function given by f(y) = ...
0
votes
1answer
51 views

Transformations

The length of time that a machine operates without failure is denoted by X and the length of time to repair a failure is denoted by Y. After a repair is made, the machine is assumed to operate like a ...
1
vote
1answer
43 views

Transformations problem

$$ \mbox{Let}\ x\ \mbox{have pdf}\quad {\rm f}\left(x\right) = {n \choose x}p^{x}\left(1 - p\right)^{n-x} $$ for $x = 0,1,2,\ldots,n$ where $n$ is positive integer constant and $0 < p < 1$ is ...
1
vote
1answer
55 views

How do i determine the charasteristic function of X^2?

I'm wondering how I kind show that the charasteristic function of $X^2$ given that $X\in N(0,1)$ is $\varphi_{X^2}(t)=\frac{1}{\sqrt{1-2it}}$. I have tried using the change of variables such that ...
1
vote
0answers
34 views

Relation with $F$ distribution and $t$ distribution

If $X\sim F_{n,n}$ , then show that $$\frac{\sqrt n(\sqrt X-\frac{1}{\sqrt X})}{2}\sim t_n$$
2
votes
0answers
58 views

Stabilize Variance for Statistics (Transformation)

Problem: When $Y (> 0)$ has mean and variance equal to $\mu$ and $\mu/n$ respectively, it is shown in the textbook that the appropriate transformation of Y to stabilize variance is the square root ...
2
votes
3answers
54 views

X has pdf $f(x) = \frac{x^{2}}{18}$ for -3<x<3, what is the pdf of $X^{2}$

So this was my solution: Say, $Z = X^{2}$, then $X=\pm \sqrt{Z}$ and, $$P(Z=z)=P(X = \sqrt{z}) + P(X = -\sqrt{z}) = \frac{z}{18} + \frac{z}{18} = \frac{z}{9}$$ for $$0<z<9$$ However: ...
1
vote
1answer
226 views

How to deal with non random data in statistical analysis?

I have a set of monthly water quality data, and I want to use them in a few statistical analysis (such as finding distribution or using in copula models) which require random variables as input. I ...
3
votes
2answers
494 views

Kernel density estimation for heavy-tailed distributions using the champernowne transformation

I am trying to follow this paper to estimate the density for a heavy-tailed distributions using the champernowne transformation. Alternative link to the paper Another alternative link to the paper ...
1
vote
1answer
551 views

Justification for transforming explanatory variables

I am using linear and generalised linear models, and have transformed my explanatory variables using $log10(\bullet)$ and $sqrt(\bullet)$ transformations, and my response variable using an arcsine ...
0
votes
1answer
6k views

Arcsine squareroot transformation for data ranging from -$1$ to $1$

According to the Handbook of Biological Statistics, the arcsine squareroot transformation is used for proportional data, constrained at $-1$ and $1$. However, when I use ...