Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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0
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1answer
31 views

How to find the variance of X in this case?

Let Y be an exponential random variable with mean $\frac{1}{\theta}$, where $\theta>0$. The conditional distribution of X given y has Poisson distribution with mean y. Then, what is the variance of ...
0
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0answers
10 views

creating matrix normal distribution

would you please help me? I have a distribution on $vec(A)$ as below $$ q(vec(A))=N_{np}(A|vec(\mu),Q) $$ In which $N(.)$ means the normal distribution and $$ Q=L\otimes K+ H\otimes J $$ How can ...
1
vote
1answer
40 views

Bhattacharya Distance (or A Measure of Similarity) — On Matrices with Different Dimensions

We have a series of observations of different properties (such as heart rate or blood sugar level and others as well) across different days from different people from different geographical regions. ...
-2
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1answer
41 views

Related to order statistics in probability theory

If $x$ and $y$ are two independently and identically distributed random variables then can we write the following? ...
0
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0answers
38 views

Correlation between $X$ and $Y$ where $X$,$Y$ are from i.i.d. standard normals and $X+Y > 0$?

Suppose $Z = X + Y$, where $X$ and $Y$ are independent standard normal random variables. If we generate plenty of $(X,Y)$ pairs and only keep the ones where $Z>0$, what's the correlation between ...
1
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1answer
31 views

Why does $P (Z=0)=\frac 1 2$ imply Z isn't normally distributed?

I understand the Wikipedia proof, except for the fact highlighted in green. $X+Y$ needs to be a normal distribution for the random variable to be normally distributed. But why does $P (X+Y=0)=\frac 1 ...
0
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0answers
57 views

How to identify what probability distribution this is?

in the context of a very long derivation (variational inference), by grouping the constant terms, I arrive at the following form that should be a probability distribution: $$ p(x) = C x^{-a} ...
-1
votes
2answers
40 views

Finding out the constant in p.d.f. with given mean?

Probability density function $$f(x)=\alpha\ e^{-x^2-\beta\ x},\ -\infty<x<\infty$$ Also $E(X)=-\frac{1}{2}$ I tried solving it using respective formulas of total probability equal 1 and the ...
2
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0answers
26 views

expectation half-normal distribution or expectation Truncated Normal Distribution [duplicate]

I want to calculate integrals $$ \begin{split} \int_0^\infty x \exp\{ ax-b x^2\} dx &= \int_0^\infty x\exp\{-b(x^2-\frac{a}{b}x)\}dx\\ &= ...
1
vote
1answer
74 views

I know the following integral can be computed in closed form, but I can't figure out how …

The following integral comes up for me when I'm computing a normalizing constant for a probability distribution: $$\int_0^\infty ...
3
votes
2answers
63 views

Fourier transform of a Lévy density $\frac{1}{\sqrt{2\pi }}\int_{0}^{\infty} e^{ikx-\frac{1}{2x}}x^{-\frac{3}{2}}dx$

A Lévy density is defined as $$q(x;1/2,1)=\frac{1}{\sqrt{2\pi }}e^{-\frac{1}{2x}}x^{-\frac{3}{2}}$$ for $x>0$ I am looking for it's Fourier transform: $$g(k;1/2,1)=\frac{1}{\sqrt{2\pi ...
1
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0answers
20 views

How to show a Bayes Update is Normal Economics Example

I'm having trouble with an example of Bayesian updating in economics. The problem I'm trying to solve is from Sargent and Ljungvist's Recursive Macroeconomic Theory, Chapter 6. A firm wishes to ...
2
votes
3answers
221 views

Confusion about the range of the sum of i.i.d. random variables

Let $X_1, X_2, ...X_n$ be independent and uniformly distributed random variables on the interval $[0,1]$. Now suppose I wanted to calculate the probability density function of $Z = X_1 + X_2 + ... + ...
0
votes
1answer
15 views

Unconditional Variance of Normal RV with mean being a NRV

I am trying to find the variance of $X$ which is defined like this: $$X \sim N(Y,e)$$ where $Y$ is a normal random variable with the distribution $Y \sim N(a,b)$. $a$,$b$, and $e$ are known ...
2
votes
0answers
27 views

Uniformly distributed random vector between the $x$ axis and a function of $x$

Let the random variable $(X,Y)$ be uniformly distributed in the domain between the $x$ axis and $y=e^{-x^2/2}$. I have to find whether $X$ and $Y$ are normally distributed random varibales, but first ...
0
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0answers
13 views

Find probability distribution to find minimum distance between four RV.

Consider, we have four random variables $X, Y, -X $ and $-Y$, where $X$ and $Y$ are circularly symmetric complex normal random variables. Now let four distances $D_1=dist(X,-X), D_2=dist(X,Y), ...
4
votes
1answer
112 views

Expected Value of Maximum of Two Lognormal Random Variables

We have two random variables $X$ and $Y$ which are log normally distributed, with suitable parameters, what is the expected value for $\max(X,Y)$? Given, $$ X=e^{\mu+\sigma Z_{1}};\quad ...
0
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0answers
48 views

comulative distribution function of z=x if x<1 z=Y+1 if x>1

So here is the question $X\sim\mathrm{exp}(1)$ $Y\sim\mathrm{exp}(2)$ $Z=X$ if $x\le1$. $Z=Y+1$ else. What is the cumulative distribution function of $Z$? So what i thought. ...
1
vote
1answer
44 views

Choosing the distribution

If in an experiment I have recorded the number of people, lets say $X$, alive at some time ($>0$), out of a sample of $n$ people, which is the best distribution for $X$ to use? The survival time ...
0
votes
1answer
25 views

Inverse CDF of a function

I am trying to find the Inverse CDF (quantile function) of this function to create an random number generator: $f(p_a) = (\beta + 1) p_a^{\beta} \text{ where } \beta \geq 1 \text{ and } 0 \leq p_a ...
1
vote
1answer
21 views

Sum of two standard variables with joint bivariate distribution?

Let $X_1$ and $X_2$ have standard normal distribution and let $(X_1,X_2)$ have a joint bivariate distribution. Can anyone explain why: $X_1+X_2=\sqrt{2+2\rho}Z$ where $Z\sim N(0,1)$ and $\rho$ is the ...
0
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0answers
9 views

Extracting Expected values with conditional probabilities from a mass probability function

I have to calculate $E(X^3,Y^4,Z)$ $cov(X,Y)$ $E(var(Z|X))$ $cov(E(Z|X),E(X|Z))$ For calculating the 1. one: I think i should do $E(X)E(X)E(X)E(Y)E(Y)E(Y)E(Y)E(Z)$ being ...
0
votes
1answer
40 views

The prob. distribution of sum of two independent random variables

I have two Random variables $X,Y$, They are independent. In which, $X,Y$ follows same distribution $$P(X=1)=P(Y=1)=0.1$$ $$P(X=2)=P(Y=2)=0.4$$ $$P(X=4)=P(Y=4)=0.3$$ $$P(X=10)=P(Y=10)=0.2$$ How ...
2
votes
1answer
46 views

Probability with flipping the coins

I flip a coin for $N$ times. I stop the flipping until I get 4 consecutive heads. Let $X=P(N\leq6)$. On the other hand, I flip the coin for exactly 6 times. Once I finish all the flips, I check ...
2
votes
0answers
25 views

Empirical Distribution converges to the distribution in $L^1$?

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d random variables with distribution $F$. Then, for each $x \in R$, do the random variables $$ F_n(x):=\frac{1}{n}\sum_{i=1}^n 1_{[X_k \le x]} $$ ...
0
votes
1answer
19 views

Probablity distribution for two particles to decay?

Let us say I have the probability distribution of the decay of one particle as: $$f(t)=\frac{1}{\tau}e^{-\frac{t}{\tau}}$$ Then how would I find the probablity distribution for the time it takes two ...
2
votes
2answers
52 views

Find prob. that only select red balls from $n$ (red+blue) balls

There are 4 blue balls and 6 red balls(total 10 balls). $X$ is a random variable of the number of selected balls(without replacement), in which $$P(X=1)=0.1$$ $$P(X=2)=0.5$$ $$P(X=3)=0.2$$ ...
0
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0answers
9 views

Distribution for frequency data

I am conducting statistical tests on a large set of frequency data. The frequency data is in the form of times of occurrence of events at a number of locations. Locations can be assumed to be ...
1
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0answers
22 views

Distribution Poisson process [closed]

A mass emit particles by a Poisson process ( about a half cup of $\mu$ particles per second). A person place a particle counter with the mass , where each particle emitted has a chance to reach the ...
1
vote
2answers
42 views

Maximum Likeliehood Question (Bernoulli or Binomial p.d.f.?)

Bernoulli Binomial I'm calculating the Maximum Likelihood for parameter p which is the proportion of purchases made by women. Since there are ...
0
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0answers
32 views

Obtaining cdf from a pdf containing absolute value and sign function

I want to calculate the cdf of the following pdf: $f(y|\mu, \sigma, k, \lambda) = \frac{C}{\sigma} exp(-\frac{1}{[1+sign(y-\mu+\delta\sigma)\lambda]^k \theta^k \sigma^k}|y-\mu+\delta\sigma|^k),$ ...
0
votes
1answer
27 views

How to find the probability distribution function from the Moment generating function?

Given that you know the moment distribution function $M_{x}(t)$, for example, $M_{x}(t)=a\exp(b\,t)$, is it possible to define the probability distribution function $f(x)$?
1
vote
1answer
30 views

Joint probability distribution function of W=Y-X

Let $X$ and $Y$ denote the arrival times of the first two calls at a telephone switch.the joint pdf of $X$ and $Y$ is: $$f_{X},_{Y}(x,y)=\begin{cases} \lambda ^{2}e^{-\lambda y} & \text{ ...
0
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0answers
61 views

Random distribution with angles for solar cell arrays.

Currently I'm stuck at a problem where I wonder if I'm not actually over thinking it. Say I have a surface (solar cell), this surface gives me power based on a reference and the angle of incidence on ...
1
vote
1answer
58 views

Compound Distribution — Uniform Distribution with Normally Distributed Parameters

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Uniform Distribution whose parameters are distributed ...
2
votes
0answers
44 views

Integrating sine with Monte Carlo / Metropolis algorithm

I'm learning Monte Carlo / Metropolis algorithm, so I made up a simple question and write some code to see if I really understand it. The question is simple: integrating sine over 0 to PI. The ...
0
votes
1answer
30 views

Biased coin toss

Let $p$, $q$ be values in $[0,1]$ and $\alpha \in [0,1]$. Assume $\alpha$ and $q$ known, and that $p$ is unknown parameter we would like to estimate. A coin is tossed n times, resulting in the ...
1
vote
1answer
31 views

Let $(X_1 ,Y_1),(X_2 ,Y_2),…,(X_n ,Y_n)$ be a sample from the uniform distribution on a disc $X^2 + Y^2 \leq \theta$, where $\theta$ is unknown.

Let $(X_1 ,Y_1),(X_2 ,Y_2),\ldots,(X_n ,Y_n)$ be a sample from the uniform distribution on a disc $X^2 + Y^2 \leq \theta$, where $\theta$ is unknown. That is, the joint density function of $(X,Y)$ is ...
0
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0answers
26 views

3 urns; numbered balls; pick urn, draw 2 balls no replacement.

i've been struggling with this one for a while now... we've got 3 urns: A,B,C A - 10 numbered {1,..,10} white balls. B - 5 numbered {1,..,5} red balls. C - 15 numbered {1,..,15} blue balls. Pick ...
1
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0answers
50 views

What mathematical background do I need to know before study this books?

These are the textbooks adopted by my university in the course "probability": Probability Theory: The Logic of Science by E. T. Jayn An Introduction to Probability Theory and its Applications, by ...
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0answers
14 views

Distribution of the norm of uniform random unit vector after linear transformation

Suppose that $\mathbf{u}$ is a uniform unit vector. It is obtained as $\mathbf{u}=\frac{\mathbf{n}}{||\mathbf{n}||}$ where $\mathbf{n}$ is a white Gaussian vector. Clearly we have ...
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0answers
21 views

Is there an older term for “Conflation” (of Probability Distributions)?

There is this paper, "Conflations of Probability Distributions", by Theodore P. Hill, from 2008. In its core, it talks about something pretty common: Mashing together some probability distributions ...
0
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1answer
22 views

Sum of Dice and Coin Problem

You have six balanced coins and two balanced dice tossed altogether. If Z is the sum of the scores on the dice and the number of heads, what is the variance and mean of Z? My attempt: I ...
0
votes
1answer
24 views

Distribution function of $\sin X$ , where $X$ is a random variable

Let $X$ be a random variable with a continuous distribution function $F$. Find expression of the distribution of $\sin X$. The solution to this problem says that if $-1 \le y \le 1$, $P(\sin X ...
0
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0answers
16 views

Finding probability density function (PDF) from autocorrelation function

Given an autocorrelation function for a random process $R_{XX}(\tau)$, how can I find the first-order probability density function $f_{X}(x)$ for this process?
0
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0answers
59 views

Hypergeometric distr. with 3 balls

We have 13 black balls, 3 red balls and 16 white balls. We want to calculate the probability for getting 16 balls where we want only the white and red balls i.e. zero black balls. We draw without ...
1
vote
1answer
76 views

Expected Value of Maximum of Two Lognormal Random Variables with One Source of Randomness

We have two random variables $X$ and $Y$ which are log normally distributed, with suitable parameters, what is the expected value for $\max(X,Y)$? Given, $$ X=e^{\mu+\sigma Z};\quad Y=e^{\nu+\tau ...
0
votes
1answer
25 views

How to find the best rotation matrix between two Gaussian random variables?

My question is really simple, given two paired sets of points $\{x_i\}$ and $\{y_i\}$ defined in an N-dimensional space $\{(x_1,y_1), (x_2,y_2), ..., (x_n,y_n)\} \in {\rm I\!R}^N \times {\rm I\!R}^N ...
1
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0answers
24 views

What are the differences between large deviations theory & extreme value theory?

I need to study both for my Master's thesis in finance. (Probably, I'll have to apply them on the Value at Risk and Conditional Value at Risk estimation, so, on quantile estimation, loosely speaking; ...
0
votes
1answer
33 views

Definition and use of Empirical Cumulative Distribution Function (ECDF)

Let $X_1 , X_2, \ldots ,X_n$ be independent identically distributed random variables with a common cdf $F(t)$. Then the empirical cdf is defined as , $$F_n(t) = \frac { \text{number of elements in ...