Questions tagged [transform]

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31 views

What is the distribution of bit counts of a binomial random variable?

Suppose I have a binomial random variable $X \sim B(n,p)$ and I apply the following bit counting operation $$Y = \operatorname{bit\_count}(X)$$ where $\operatorname{bit\_count}$ is defined in the ...
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0 votes
1 answer
17 views

Do linear transforms of IID random vectors make the resulting covariances sufficient to describe statistical dependence?

This question is motivated by the fact that a linear transformation of an IID standard normal vector gives the multivariate Gaussian distribution, and that the statistical dependence of such ...
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1 vote
0 answers
24 views

Deriving distribution under change of variables between spaces of unequal dimension

For a function of random variables $T:\mathbb{R}^n \mapsto \mathbb{R}^m$ Wikipedia outlines how to handle three cases: $m = n = 1$ $m=n > 1$ $n>1 \land m=1$ There seems to be two missing cases:...
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0 votes
0 answers
24 views

Complicated Transform of a Beta Random Variable

Consider a Beta distributed random variable, $X \sim Beta(a,b)$ Then consider the transform $$Y = \sqrt{K\frac{X}{1-X}}$$ where $K > 0$ is a constant. How could you go about finding the PDF of $Y$?...
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2 votes
0 answers
17 views

Distribution of the Square Root of a Beta Prime Random Variable

Given a Beta Prime distributed random variable $X \sim BP(a,b) $ with probability density $$\rho_X(x) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\frac{x^{\alpha - 1}}{(1+x)^{a+b}}, x > 0$$ Consider ...
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0 votes
0 answers
28 views

Scale of X axis in forest plot

The question is very simple. For the help file of the meta package, let's consider this forest plot: ...
6 votes
1 answer
56 views

What is the distribution of a random linear combination of gamma random variables?

Let $U\sim \mathcal U(0, 1)$ be a random variable uniformly distributed over the interval $[0, 1]$. Let $X_1, X_2\sim \Gamma(a, b)$ be two iid random variables with a Gamma distribution. Now it is ...
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0 votes
0 answers
17 views

log transformation with the self-reported health score variables

With self-reported health scores (as a dependent variable) - such as CES-D for depression which typically ranges from 0 to 60, I was wondering whether log-transforming these scores would allow me to ...
0 votes
0 answers
54 views

How does skewness in the independent features affect the performance of classification algorithms?

Do we need to take action when we encounter a skewed feature (not the target)? We can log transform for regression etc., but does it matter in also classification models?
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10 votes
3 answers
1k views

If a data set appears to be normal after some transformation is applied, is it really normal?

Suppose you have a data set that doesn't appear to be normal when its distribution is first plotted (e.g., it's qqplot is curved). If after some kind of transformation is applied (e.g., log, square ...
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0 votes
0 answers
23 views

Finding density function of random vector (density transformation)

Let $X_{i} \sim \operatorname{Ga}\left(\alpha_{i}, \lambda\right)$ independent with $\alpha_{i}>0$ fūr $i=1,2$. Furthermore it is known, that $V=X_{1}+X_{2} \sim \operatorname{Ga}\left(\alpha_{1}+\...
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0 votes
0 answers
451 views

Transforming only y variable with Yeo-Johnson in Python

I have 4500 values in my dependent variable and 75% of values are zeros (no negative ones) The distribution looks like this. In multiple sources I read that Yeo-Johnson transformation can be a ...
0 votes
0 answers
128 views

Prior after variable transformation

I am sampling the variables ($\alpha, \beta, \gamma$) with a Markov chain where I put the priors to be flat, e.g. $\pi(\alpha) = U(\alpha)$ and similarly for $\beta, \gamma$. Now I am actually ...
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4 votes
1 answer
237 views

What is the expected value of the log-log-normal distribution (aka the LLN distribution)?

Question 387180 discusses the pdf of the log-log-normal distribution. I'd like to know if there's an expression for the mean of this distribution. I'm trying to work it out with pencil and paper, but ...
2 votes
1 answer
209 views

Anscombe transform vs log transform

I am trying to understand in layman's terms how the anscombe transform converts a poisson distribution into a normal distribution. So, why is a log transform not sufficient in its own to obtain the ...
2 votes
4 answers
132 views

Approximate distribution of a complicated function of a random variable

If $X$ is a random variable cdf $F(x)$ such that $F$ is invertible then we have the standard method of finding the pdf of any function of $X$, say, $\sin(X) $ or $ X^3+1 $.However,in many situations ...
1 vote
1 answer
29 views

Integrating in log-space with a change of variable

I have a probability density function $P(f|\mu,\sigma) = \mathcal{N}(f|\mu,\sigma)$. I need to change the variable $f$ to $L = \log_{10}[f]$ so I can integrate it jointly with another PDF whose domain ...
0 votes
0 answers
29 views

How to convert a normal random variable to a truncated normal distribution? [duplicate]

Is it possible to transform a normally distributed variable into one that defined by a truncated normal distribution? I am currently using a KL transform to generate Gaussian random fields. I would ...
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0 votes
1 answer
32 views

How can I fix the residual plot though transformation

What transformation can I use to fix this residual plot (make the red line horizontal). I tried square root, log, 1/y, and squared. None of them helped. The data is of a 2 way ANOVA: Response ...
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7 votes
1 answer
255 views

Distribution of i.i.d. random variables $X$ and $Y$ if $XY \sim \text{Beta}(\alpha, \beta)$

This is a follow-up to the question Square root of a Beta(1,1) random variable, which received two great answers. If $XY \sim \text{Beta}(\alpha, \beta)$, and $X$ and $Y$ are two independent ...
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11 votes
2 answers
881 views

Square root of a Beta(1,1) random variable

If $X^2 \sim \text{Beta}(1,1)$, is there a closed form for the distribution of $X$? If yes, what does it look like? And if this is not too much to ask, is there a general way to find the distribution ...
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0 votes
0 answers
38 views

Can I transform percentage data using just the square root?

I have a data set with the number of larvae (out of 100) that have metamorphosed after 0,1,2, and 5 days. I want to perform a repetitive ANOVA on the results but they are not normally distributed. I'...
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-1 votes
1 answer
102 views

Calculation of integrals transforming $N(μ,σ^2)$ to $N(0,1)$

Let's say $X\sim N(\mu,\sigma^2)$, where $\mu$ and $\sigma^2$ are known. How can we calcuate the following integrals by transforming $X$ to $Z\sim N(0,1)$? $$ \int_{c_1}^{c_2}(x-c)\frac{1}{\sigma\sqrt{...
0 votes
0 answers
18 views

Confusion in the domain of transformation of random variables

I have two random variables $X$ and $Y$. Let us assume both ranges from $0<x<\infty$ and $0<y<\infty$. Let is also assume both are having exponential densities. I am making two different ...
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0 votes
1 answer
38 views

What distribution is this? [duplicate]

Basically, I am told that $\varepsilon$~$N(0,1)$, and $\omega$~$IG(\frac{v}{2}$,$\frac{v}{2})$ where $IG$ is the inverted gamma distribution Now, I am told that the distribution of: $\varepsilon(\frac{...
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1 vote
2 answers
1k views

Finding the pdf of Y from that of X, linear transformation

The question is Let $X$ be a continuous random variable with pdf $f_X(x) = 2(1 − x)$, $0 ≤ x ≤ 1$. If $Y = 2X − 1$, find the pdf of $Y$. I understand these steps$$F_Y(Y ≤ y) = P(2X-1 ≤ y) = P(X ≤ (y+...
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3 votes
1 answer
363 views

Z transform of $ 2^{-|n|} $

Dears, I'm trying to compute the Z-transform of $$ x(n) = 2^{-|n|} $$. My procedure is as follows: Using definition of Z transform: $$ X(z) = \sum_{n=-\infty}^{\infty}2^{-|n|} z^{-n} = \sum_{n=-\infty}...
0 votes
0 answers
29 views

Transforming Logistic Regression Model

I have a Logistic Regression Model in R ...
0 votes
1 answer
110 views

Copula between a distribution and its univariate transformation

I'm trying to compute the copula (or joint distribution) between x and a univariate transformation, like say sin(x). That is compute $C_{XY}$ (or $F_{XY}$) given that $x \sim U(0,1)$ and $y = sin(x)$ ...
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6 votes
1 answer
452 views

Inverse Gaussian chi square connection

The inverse Gaussian distribution $IG(\mu,\lambda)$ is associated with the density $$f(x;\mu,\lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}\,\exp\left\{-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right\}\qquad \...
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1 vote
1 answer
142 views

PMF of $aX_1 + bX_2$ (Bernoulli)

Let $Y_1 = aX_1 \sim \text{Bernoulli}(p)$ and $Y_2 = bX_2 \sim \text{Bernoulli}(p)$, what is the PMF of $Z = Y_1 + Y_2$ for $a > 0$, $b > 0$ and $a \neq b$? Can somebody check my result? $$p_{...
-1 votes
1 answer
68 views

A Normal distribution variable in the power of N

If $X$ is normal distributed random variable, what is the distribution of $|X|^n$? I am struggling to understand the distribution. Any guidance would be appreciated.
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8 votes
1 answer
2k views

Why can I interpret a log transformed dependent variable in terms of percent change in linear regression?

Looking at resources such as this one and this one, you see claims like "Exponentiate the coefficient, subtract one from this number, and multiply by 100. This gives the percent increase (or ...
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2 votes
2 answers
614 views

How do you get the double sum or integral from $E(X+Y)$ (expected value)?

I was given a proof for $E(X+Y)$ = $E(X)+E(Y)$ for cases where both variables are either discrete or continuous: Discrete: $$ \begin{align*} E(X+Y) &=\sum_{x\in\mathcal X}\sum_{y\in\mathcal Y}(x+y)...
1 vote
0 answers
84 views

Laplace-Stieltjes Transforms and distribution

I was going through a paper, I came across below relation, \begin{equation} T=\begin{cases} C, & \text{with probability $P(H<C)$}\\ 0, & \text{with probability $P(H>C)$} \...
1 vote
0 answers
126 views

Is the copula function invariant only under deterministic monotonic transformation?

I read about the following theorem (see Proposition 3 in the picture below) on the invariance of copula under monotonic transformation, my questions is: 1. Are the $T_i$ mentioned in the following ...
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3 votes
1 answer
361 views

Change of variables in pdf

I have the joint pdf$$f(x_1,x_2)=x_1e^{-x_1(1+x_2)}I_{(0,\infty)}(x_1)I_{(0,\infty)}(x_2)$$and have to derive the joint pdf of $$Y_1=e^{-X_1}\qquad\text{ and }\quad Y_2=e^{-X_1X_2}$$ I set $x_1=-\ln(...
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1 vote
1 answer
66 views

How can I shift the average probability keeping constraint (0.0:1.0)?

I have a large datasets of values that range from 0 to n. I am interpreting the values as probabilities for a later pseudo-random selection process. To make the values serve as probabilities, I ...
1 vote
1 answer
125 views

How to interpret a specific data transformation?

I came across this specific data transformation in the context of a physics application, which by itself is rather complex and hence out of the scope of this question. However since this ...
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2 votes
1 answer
38 views

How do you work with a function of a uniform distribution? [closed]

I am struggling with parts b and c. How do you solve them? Could you please give the solution?
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0 votes
2 answers
152 views

how do you transform/standardise a function to always give values between y1 and y2?

Having lost some of my math skills, I am having problems with something that I think should be fairly easy but is eluding me: I have a plateau shaped function that I would like to standardise such ...
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1 vote
0 answers
52 views

Can I use Linear Regression or do I need Nonlinear Regression

I am trying to fit these two equations to data in R via regression. First Equation: $$y(x) = a + \frac{b}{c + x^m}.$$ This equation is constant plus reciprocal function, resulting in a hyperbolic ...
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0 votes
1 answer
308 views

Normalize target value for linear regression

I'm building a regression model to predict sensor value over time. Bellow is a figure of my sensors data over time: Based on this video about transforming nonlinear data with a log function, What ...
0 votes
0 answers
124 views

Squeeze a time series to fit in a range while maintaining shape

I have the following time series with intermediate highs and lows marked by the vertical lines: I want to transform/squeeze the series so that the resulting series would fit in a range, let's say [0,...
4 votes
1 answer
558 views

What is the general second-order Taylor approximation to $\mathbb{V}(f(X))$?

If $X \sim \text{N}(0, \sigma^2)$ it is well-known that we have the second-order Taylor approximation: $$\mathbb{V}[f(X)] \approx f'(\mu)^2 \cdot \sigma^2 + \frac{f''(\mu)^2}{2} \cdot \sigma^4.$$ ...
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1 vote
1 answer
1k views

How to present Confidence Interval for Log-Transformed Means & Mean Difference?

After trying to read on this topic, I still have some clarifications remaining. Context: Comparing between 2 arms (categorical), measuring microbiological plate-counted bacteria concentration (...
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0 votes
1 answer
3k views

How to reduce kurtosis of data

I'm trying to reduce the kurtosis of my dataset and make it approximately Gaussian, with a common-sense uni-modal shape. The raw data looks like this: I first tried ...
1 vote
0 answers
71 views

What is the laplace transform of the below given PDF?

Really am interesting to know more about statistical properties of the following PDF , of the Random variable $z$: $$F(\sigma,\mu,z)= \frac{(z-\sigma )^2 \exp \left(-\frac{(z-\sigma )^2 \sqrt{\left(...
1 vote
0 answers
44 views

How to transform/convert likelihoods to scores?

I have the probability of loan default for a labeled dataset where the distribution of probabilities is heavily skewed. Labels are defined as "good/0" for no default and "bad/1" for defaults. My goal ...
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2 votes
1 answer
232 views

Transformation of Confidence Interval = Confidence Interval of Transformation? [duplicate]

I am wondering about the following situation: I have a confidence interval estimator $\delta(x)=[lb, ub]$, which returns valid a%-confidence intervals for a value $\theta \in \mathbb{R}$ (not ...
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