0
votes
0answers
76 views

Transformation of variables

Let variables $U$ and $V$ be uniformly distributed on $[-\pi, \pi]$, and independent. Let: $$(x,y) = (\cos(U+V),\sin(U-V))$$ What is the probability distribution function of $f_{x,y}(x,y)$ My ...
0
votes
2answers
21 views

Strongest 'average' for a diverse set of numbers?

I have a set of numbers consisting of two general size numbers: size 'a', and size 'b' which are about three times bigger in size than size 'a'. There is some variation and the list might look like ...
1
vote
3answers
66 views

Transformation of two independent uniform random variables

Suppose $X,Y \sim \text{Uniform} \left(0,1 \right)$ are independent. Then I need to find the PDF for $W=X/Y$. By the CDF technique this is seen to be : $$F_W( w)=\int_{0}^1 \int_{0}^{wy} ...
0
votes
0answers
51 views

More on transformations and convolution on continuous random variables

This question is related to my last question but I've done some more exploring and then got stuck again. I decided to modify the problem a little bit and use a transformation of a random variable that ...
1
vote
1answer
58 views

Looking for peculiar vector transformation

I have a vector of numbers from 0 to 1. For example: [0.5, 0.5, 0.1]. I need to find a transformation which increases sum of the vector to asked number and: -keeps the order of elements (if element1 ...
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 ...
1
vote
0answers
47 views

One to one Bivariate Transformation

Why does the below show the transformation is one to one? These are lecture notes ( the text and the blue writing)
-1
votes
1answer
52 views

What's the pdf of $Z=X^2 +2X$ if $X$ is a standard normal? [closed]

Le be $X$ distributed as a standard normal. What is the density function of $Z=X^2 +2X$? Thanks for your help
0
votes
0answers
19 views

Differential probabilities of transformed random variables

Let $X$ be a random variable with probability density function (pdf) $f(x)$ and let $Y = h(X)$ be a transformation on rv $X$. While calculating the pdf of $Y$, it is assumed that $Prob(x \le X \le x + ...
0
votes
1answer
47 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 ...
1
vote
0answers
32 views

Manipulating this probability distribution function

I have a probability distribution function as follows: $$ P(y|x,w, \phi) = \frac{\phi}{2\pi} \exp ^{-0.5 (y-t(x, w)'\phi (y-t(x,w)) } $$ Here $y$ and $x$ are two observed values. $\phi$ is also some ...
1
vote
1answer
75 views

Expectation of (1/x)-1 possible transformation involved??

I'm a bit confused with the first steps in this problem: $F(x)=x^4$ for $0<x<1$ a) Find $E[(1/X)-1]$ b) Let $Y=(1/X)-1$. Find the support of $Y$, its pdf and CDF. Name its ...
0
votes
0answers
96 views

Conversion of covariance matrix from Cartesian to Spherical coordinates for integration

I have to perform a convolution of a function in polar coordinates $\rho(\textbf{x}) = \rho(r,\theta,\phi)$ with a function $P(\textbf{x}) = P(x,y,z)$ in cartesian coordinates. $\int ...
0
votes
0answers
28 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
0answers
29 views

Density of transformation of normal distribution

A data set contains real values $\left\{v_1,v_2,\text{...},v_k\right\}$, $k<\infty$. $X_n\sim \mathcal{N}(\mu ,\sigma ),\ n=1,2,...,k$ $P$ is the (not necessarily unique) permutation that ...
1
vote
0answers
28 views

$f_{X^2}(x)$ VS $f_X(x^2)$ [duplicate]

Sorry, this time the format should be accurate. In probability, when we try to describe a pdf, we write it as $f_X(x)=1/x$, which means the random variable is X and the x is the specific variable in ...
0
votes
1answer
78 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
45 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
37 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
2answers
37 views

Given $f_{X,Y}(x,y)$, what is the pdf of $Z=XY$?

I approached the problem as following: $f_{Z}(z) = \int_{0}^{\infty} f_{X,Y}(X=\frac{z}{y},Y=y)dy$ However, according to the textbook, the problem should be solved as below: $f_{Z}(z) = ...
1
vote
1answer
54 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
30 views

Relation with $F$ distribution and $t$ distribution

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

$t$ distribution

Let $X$ and $Y$ be iid random variables with $t$ distribution with $n$ degrees of freedom ,$t_n$. Show that , $$\frac{\sqrt n(Y-X)}{2\sqrt{XY}}$$ also follows $t_n$ distribution
0
votes
0answers
94 views

check the independence of transformed variable of two independent Gamma random variables

Let $X$ and $Y$ are two independent random variables following Gamma Distribution $X\sim \Gamma(\alpha,0,1)$ and $Y\sim \Gamma(\beta,0,1)$ Show that the ...
2
votes
0answers
195 views

Derive Student T distribution using transformation theorem

I am trying working on an exercise that asks me to show that If $ X_1 \in N(0,1) $ and $ X_2 \in \chi^2(n) $ are independent random variables, then $ X_1 / \sqrt{X_2/n} \in t(n) \, $ where $ ...
2
votes
3answers
53 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: ...
3
votes
2answers
461 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 ...
0
votes
1answer
106 views

Legendre transform of log moment function

Here is something I do not understand for my lecture notes. The lemma is this. Let $\mu$ be a probability measure on $R$, and $\Lambda^*_\mu$ is the Legendre transform of $\mu$. $\Lambda_\mu^*\geq ...
0
votes
2answers
58 views

Multivariate Random Variables

$f(x,y) = {2\over 5}(2x+3y) \quad for\quad 0<x,y<1 $ and we want to know the distribution of $2X+3Y$ I did it in a very lousy way which is let $ U=2X+3Y ,\; V=X$ Then have $\;f_{U,V}(u,v)$ ...
1
vote
1answer
71 views

Transforming a Continuous Function

My math is quite limited so please bear with me. I will get to the point: Is there a way to transform a continuous function into a bounded one? In essence I have a normalized Gaussian distribution ...
1
vote
2answers
172 views

Modifying a discrete probability distribution according to set of weights

Given a discrete probability distribution (e.g., ${P_1=0.85,P_2=0.05,P_3=0.05,P_4=0.05}$), I would like to transform it according to some set of "weights" (say, ${w_1=2,w_2=0.5,w_3=1,w_4=0.5}$), which ...
2
votes
1answer
90 views

random variable transformation

I'm having trouble with the following random variable transformation: $Y = X^2 + X$ I am looking for the pdf of Y. I tried the following method: $p_Y(y) = \int_{X} p_{Y|X=x}(y)\cdot p_{X}(x)dx$ and ...