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

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0answers
52 views

Generating Random Number From Discontinuous Distribution Function

I have the following density function : I want to generate random variate from this density function according to acceptance-rejection algorithm , Inverse transform algorithm and composition ...
2
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1answer
1k views

Probability of finding P(X=k)?

A factory produces 10 glass containers daily. It may be assumed that there is a constant probability p=0.1 of producing a defective container. Before these containers are stored they are inspected and ...
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0answers
86 views

multivariate non-negative distributions

I need a multivariate distribution defined over any vector (not special matrices as in case of Wishart distribution etc.) whose elements are non-negative (must include 0 as well). There is no other ...
2
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1answer
173 views

Using Central Limit Theorem when we NON-IID sample

I'm trying to solve a CLT question and I've got some issues. I appreciate if you could help me on that. Consider the question below: $\epsilon_i$ 's are iid random variables with finite mean and ...
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2answers
64 views

What do I need to study to do this Gaussian question?

I'm taking a probabilistic machine learning course and need to understand some background mathematics, including the following question: Let $x$ be a Gaussian random variable with mean $μ$ and ...
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1answer
37 views

Computation of an integral

While computing densities for some distributions, I stumbled on the following family of parametrized integrals: $$ p (x) := \sqrt{\frac{2}{\pi}} \int_{\mathbb{R}_+} e^{-\frac{x^2}{2 y^2} - y^2} \ d ...
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1answer
27 views

Expected value and variance of n independent links

A chain is made by connecting n links. The lengths of different links are independent and uniform over [50,70] millimeters. What is the expected value and variance of the length of a chain obtained by ...
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0answers
63 views

How to learn mixture Gaussian with inequality constraint of component variances

Let $f_1(x)$,…,$f_n(x)$ be Gaussian density functions with different parameters, $\mu_i$ and $\sigma_i$ are the parameters (mean and variance) of the Gaussian component i, and $w_1,\ldots,w_n$ be real ...
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1answer
49 views

Marginal distribution of $P$

Joint distribution of $P$ & $Q$ is $$f_{P,Q}(p,q)=\frac{1}{2\sqrt{(2\pi)}\sigma}\exp[-\frac{1}{2}{(\frac{\frac{p+q}{2}-\mu}{\sigma})^2}] \times\theta\exp[-\theta(\frac{p-q}{2})],\quad ...
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2answers
179 views

Inverse quantile function for $\sin^2(x)$

What is the transformation which takes the standard uniform distribution $U[0,1]$ to the following probability density function $f$: $$f(x)=\sin^{2}x$$ Where $x\in\left[0,\pi\right]$ and ...
2
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1answer
88 views

A variance-mixture model

So I've tried to make a probability distribution which has a tunable degree of kurtosis and which becomes Gaussian if the control-parameter goes to 0. Now Levy-distributions are out of the question, ...
2
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1answer
19 views

A question about distributions/densities

Given two random variables $X,Y$ how to show that $P(X\leq Y+x)=\int F_X(y+x)f_Y(y)dy$? I know that $f_Y(y) = \int f_{XY}(x,y)dx$, but have no idea how to go with the previous equation.
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3answers
126 views

If $X\sim \exp(\lambda)$ and $Y\sim \exp(\mu)$ then $P(X\leq Y)=\frac{\lambda}{\lambda+\mu}$. Is there an intuitive interpretation for this fact?

I can verify this via double integrals, but I'm wondering if this can be put in the context of a Poisson process or something to give it an obvious meaning. I can't think of exactly how it would work. ...
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1answer
780 views

Using a Poisson distribution, how do find if 'X' or more of 'Y' trials pass?

I am presented with the questions: Flaws occur in mylar material according to a Poisson distribution with a mean of 0.05 flaw per square yard. (a) If 29 square yards are inspected, what is ...
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1answer
288 views

Are $X$ and $Y$ independent or uncorrelated? [closed]

let $Z$ be a uniformly distributed random variable over the range $[-1,1]$ let $X=Z$ and $Y=Z^2$ be random variables. a) Are $X$ and $Y$ independent? b) Are $X$ and $Y$ uncorrelated?
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2answers
931 views

How to convert a histogram to a PDF

I know this may be an easy question, but due to lack of math knowledge I do not know the answer. Would you please explain to me with a simple example that how can I find PDF from a histogram. Thank ...
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2answers
67 views

Mean And Variance Of Beta Distributions

I want to find mean and variances of beta distribution . The distributions function is as follows: when x is between 0 and 1 Searching over internet I have found the following question .Beta ...
3
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3answers
84 views

How to show that $p_{X,Y}$ cannot be derived from $p_X$ and $p_Y$?

This question is about marginal/joint distributions. Given some pdf $p_X$ and $p_Y$, how can I show that I cannot derive the joint density functino $p_{X,Y}(x,y)$ in general? I guess that it has ...
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6answers
920 views

Is it possible that two density functions that have the same mean and variance, but different distributions?

Is it possible that two density functions that have the same mean and variance, but different distributions? Can you give an example?
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1answer
2k views

Relation between Weibull and exponential distributions

The probability distribution function of a Weibull distribution is as follows: $$ f(x) = a\cdot b^{-a}x^{a-1}\cdot e^{(-x/b)^a},\quad x>0 $$ for parameters $a,b>0$. I have to show that ...
4
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3answers
2k views

Showing that ${\rm E}[X]=\sum_{k=0}^\infty P(X>k)$ for a discrete random variable

Let $X$ be a discrete random variable whose range is $0,1,2,3,\ldots$. Prove that $$ {\rm E}[X]=\sum_{k=0}^\infty P(X>k). $$ How to prove this? I tried a bit but unable to post due to formatting ...
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1answer
209 views

Mean and variance of geometric function using binomial distribution

Can anyone help solving this question please? I tried but not sure of the steps to reach the conclusion.
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1answer
48 views

probability distribution with discrete random variable

I have tried it out, but finding it difficult to post them in proper format. Please help me with the solution.
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2answers
170 views

poisson random variable distribution using probability

Let $X$ be the number of emails that a company receives in a day. Assume that $X$ is a Poisson random variable with parameter $\lambda$. The company classifies each email as spam or not spam. The ...
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0answers
2k views

Mean and variance of geometric distribution , probability

$2.$ Compute the mean and variance of the geometric distribution. Hints: It is sometimes easier to compute $\text{E}[X(X-1)]$ than $\text{E}[X^2]$. Note that ...
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1answer
44 views

probability of mass function of a discrete RV given find x probability [closed]

The probability mass function of a discrete RV X is given in the table below. Compute the following: (a) the probability X is even (b) the probability that 1 ≤ X ≤ 8 (c) the probability that X is ...
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1answer
58 views

Simulating Poisson distribution

I'm wondering if it is possible to simulate the Poisson distribution using the Alias Method, because we suppose to use this method for discrete random variables with finite support. So I think finite ...
5
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1answer
873 views

Formal definition of conditional probability

It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space $ (\Omega, \mathscr{A}, \mu ) $ ...
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1answer
2k views

Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
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1answer
454 views

Characteristic function of random variable $Z=XY$ where X and Y are independent non-standard normal random variables

I would like to find Characteristic function of random variable $Z=XY$ where X and Y are independent normal random variables, but they are not standard, i.e. $$X\sim N(\mu _x,\sigma_x)$$ $$Y\sim ...
1
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1answer
2k views

maximum of two uniform distributions

I have a question. Let's suppose that the two random variables $X1$ and $X2$ follow two Uniform distributions that are independent but have different parameters: $X1 \sim Uniform(l1, u1)$ $X2 \sim ...
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2answers
2k views

What is the PDF of random variable Z=XY?

Given two independent random variables X and Y, how can I find the PDF of random variable $Z=XY$? *If their joint distribution is required, assume that we also have it.
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3answers
92 views

Random variable $X$ inducing a distribution on $V$

I have been learning about discrete probability and found a somehow confusing (to me) definition of distribution of a random variable $X$ on a set $V$. The definition of a Random variable $X$: $$ ...
4
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1answer
1k views

How to get PDF from characteristic function

I would appreciate if anybody could explain to me with a simple example how to find PDF of a random variable from its characteristic function. Thank you.
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2answers
267 views

Expectation of Log of a Cauchy-distributed Random Variable

I found this in an article, but I cannot follow the step to get $\mathbb E[\log |a_{N,k}|]$. I'm quoting the paper: Let $a_{N,k}$ be Cauchy-distributed random variables with parameter $N(k+1)$. The ...
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1answer
257 views

Convolution of Discrete Uniform ,$DU$, Distribution.

If $X\sim DU(k,a,h),\quad -\infty<a<\infty,h>0=1,2,\ldots$ then the probability function is $$P(X=a+jh)=\frac{1}{k},\quad j=0,1,\ldots,k-1$$ Let $Z\sim DU(r,0,s)$ and $Y\sim DU(s,0,1)$ , ...
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1answer
2k views

Concept of Variance

I am curios about the concept of Variance. I try to get the better understanding of the variance by checking extreme cases. $Var(X) = E[(X^2)] - (E[X])^2$ question 1. What does it mean when Variance ...
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1answer
224 views

Conditional expected values of dependent gaussian variables

I obviously don't understand multivariate gaussian variables as well as I thought. I have k+1 variables. One which is special, call it X, and I want to find the mean of, given necessary and sufficient ...
1
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1answer
469 views

problem on random variable in probability

A game consists of first rolling an ordinary 6-sided die once and then tossing a fair coin once. The score, which consist of adding the number of spots showing on the die to the number of heads ...
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0answers
48 views

Removing density functions 'offset'

I'm trying to transform a density function (in black) to another one (in blue) in this spirit: local minima are linked together, to form a concave function in red. This function is then subtracted ...
0
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1answer
64 views

As $N\to\infty$ Hypergeometric distribution reduces to Poisson distribution

Let $X_N\sim Hg(N,\lambda N^2,N^3),\quad N=1,2,\ldots$ $$P(X_N=m)=\frac{\binom{\lambda N^2}{m}\binom{N^3}{N-m}}{\binom{\lambda N^2+N^3}{N}}$$ Now I have to show that for fixed $m=0,1,\ldots,$ ...
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0answers
41 views

Hypergeometric distribution, $Hg(1,a,b)$ follows Bernoulli with $Be(\frac{a}{a+b})$

The probability function of Hypergeometric distribution , $Hg(n,a,b)$ is $$P(X=m)=\frac{\binom{a}{m}\binom{b}{n-m}}{\binom{a+b}{n}}$$ I have to show $Hg(1,a,b)$ follows Bernoulli with ...
0
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1answer
155 views

Find expectation of function of Poisson random variable

Please, check my work. Is it correct that if $X_1,X_2,\ldots,X_n$ are independent Poisson random variables, each with a parameter $\lambda$, then $$ E\left( ...
0
votes
2answers
425 views

Shape of distribution of infinite Sum of weighted Gaussians

Let $x_i$ be samples from gaussian distribution with mean $0$ and variance $\sigma^2$ and $s_n=\sum_{i=0}^n2^ix_i$. What can one say about the distribution of $s_n$ at $n\rightarrow\infty$? Sum of ...
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0answers
25 views

mutual information for power low distributed data?

I have a dataset with power-low distribution, I would like to measure some kind of correlation/mutual information between the data to its class,(for feature selection task), Can I use ...
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1answer
581 views

Mean, mode and median equal for a random variable

Suppose for a random variable mean, mode and median are equal. What is the intuitive meaning of this ?
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1answer
61 views

Constant Density Function Property

I was wondering if, for any pdf of the type: $f_{x,y}(x,y) = c$, we can just calculate the area of integration and interpret it as the probability of the random vector. I know this would be true if ...
8
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1answer
249 views

If $X,Y,Z$ are iid unif[0,1], then $(XY)^Z \sim \text{unif}[0,1]$.

Here's a mind-blowing fact (to me at least) that is perhaps not so well-known: If $X, Y, Z$ are iid uniformly distributed in $[0,1]$, then $W = (XY)^Z$ is also uniformly distributed in $[0,1]$. If ...
3
votes
2answers
524 views

Characteristic Function of Inverse Gaussian Distribution

The pdf of Inverse Gaussian distribution, IG$(\mu,\lambda)$, is : $$p_X(x)=\sqrt\frac{\lambda}{2\pi x^3}\exp\left[\frac{-\lambda}{2\mu^2x}(x-\mu)^2\right];\quad x>0,\lambda,\mu>0$$ I have to ...
4
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1answer
54 views

Measurability of integral

Consider a function $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ which is continuous in the first argument, measurable in the second. Let $m: \mathcal{B}(\mathbb{R}^m) \rightarrow ...