Identify subset of population to match given moments

I have a Gaussian probability density function f(x;μ,σ). The mean and the standard deviation of this pdf is known. I have an unknown probability density function g(x). I only know g(x) by its n realizations {X1,X2,…,Xn}. g(x) is different from f(x) but the density functions are expected to look somehow similar. I would like to … Read more

min/max and probability distributions

We have two identically distributed, independent, uniform variables on interval [0,1] : X1, X2. And Y1=max, Y_{2}=\min(X_{1},X_{2}). I want to find distribution f(y_{1}|y_{2}) and f(y_{2}|y_{1}). So I tried to approach it like this: firstly try to calculate distribution of f(y_{1},y_{2}=w) as a derivative of: F_{Y_{1},Y_{2}}(y_{1},y_{2}=w)=\mathbb{P}([{X_{1}}\leq y_{1} \wedge X_{2} \leq y_{1}]\wedge [X_{1} \leq w\ \vee X_{2}\leq … Read more

Quantiles based on raw data or density

I’ve created violin plots of my data and included quantile lines. These were created with the geom_violin function from ggplot2 in R. However, I realized that these quantiles differed from the ones I get when I calculate them myself with the quantile function. After some investigation I found out that geom_violin seems to determine quantiles … Read more

Density Estimation of multivariate random variable

I have a dataset (300×14 matrix). This means it has 14 features and 300 observations. n=14 (a110…a1n0a22…a2n⋮⋮⋱⋮00…a300,14) Now, is it unreasonable if I estimate the density (p.d.f) of each row of this matrix (a110…a1n)separately using kernel density? Or should I calculate the p.d.f of the whole matrix? In other words, it means I want to … Read more

Algebra on random variables

I have the feeling this should be doable, or at least have an approximation, but I’m failing to find one. Let’s consider a random variable C, that belongs to a Truncated Exponential distribution between 0 and 1. If we observe n i.i.d. variables C, what is the distribution of the harmonic mean of this set? … Read more

Goodness of Fit test with PDF, not data

I have a sample probability distribution and want to know how well a Gaussian would fit to it. However, I just have a probability distribution to test, and not an actual data sample. (Nor do I want to create one, as that would be fairly computational expensive if you have several hundred of these.) How … Read more

Transforming a uniform PDF to a Gaussian PDF

I have a Uniform PDF from [-50, 50], I would like to transform it to a Gaussian. The methods that I read up about doing this(like Box Mueller) assume that the uniform distribution is between [0,1). Is there any way to do this for [-50,50]. Is there any translation or scaling that may be done … Read more

Differential histogram bin calculation

I want to be able to minimize a difference between two distributions P(x|θ) and Q. I can choose Q (e.g. to be a Gaussian normal), but P(x|θ) is an unknown distribution, so I am currently using a histogram. (It appears to be multimodal for one thing.) That is, I have some parameterized function producing data … Read more

Joint distribution of multivariate normal

Let $X$ and $Y$ be i.i.d. $N(0, 1)$, and let $S$ be a random sign (1 or -1, with equal probabilities) independent of $(X, Y)$. \begin{align*} P((SX,SY)∈B)&=P((X,Y)∈B,S=1)+P((−X,−Y)∈B,S=−1) \\ &= P((X,Y)∈B)P(S=1)+P((−X,−Y)∈B)P(S=−1)\\ &= 0.5*P((X,Y)∈B) + 0.5*P((−X,−Y)∈B)\\ &= P((X,Y)∈B) \end{align*} What does this infer about the joint distribution of $(SX, SY)$?. How does it mean that the joint … Read more