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

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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 ...
1
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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 ...
8
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1answer
970 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 ) $ ...
1
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1answer
3k 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 ...
0
votes
1answer
470 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 ...
5
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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.
0
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3answers
93 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.
3
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2answers
269 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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votes
1answer
258 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 $Y\sim DU(s,0,1)$ , ...
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 ...
0
votes
1answer
228 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
480 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 ...
1
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0answers
49 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 $m=0,1,\ldots,$ ...
0
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0answers
42 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
159 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
436 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
26 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 ...
1
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1answer
594 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 ?
0
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1answer
62 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
262 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
539 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
55 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 ...
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1answer
2k views

Finding the mean and Std Dev from Geometric probability

Wasn't sure what to call this.. .But here goes I am having trouble solving the following questions, here is the first, and how I have gone about it ...
0
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1answer
53 views

what is the probability that the first account containing substantial errors is the third one to be audited

a certified accountant has found that 9 of 10 company audits is the third one to be audited. the accountant audits a series of company accounts what is the probability that the first account ...
1
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0answers
63 views

Difference between $\frac{1}{n} \sum_{i=0}^n \frac{b_i}{a_i+b_i}$ and $\frac{\sum_{i=0}^n b_i}{\sum_{i=0}^n a_i + b_i}$

Consider scenario like this: each user can add movies. each user can rate movies he added. I want to know how many movies average user has unrated (as in average percentage). Let's translate this ...
0
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1answer
77 views

Find the two-dimensional random distribution

I need to find the two-dimensional distribution $(\xi_1,\xi_2)\in \mathbb{R}^{2}$, that for any $a,b$ the random variable $\chi = a\xi_1 + b\xi_2$ has density only if $ab\neq0$. distribution ...
0
votes
1answer
83 views

Given density $f_U(u)$ and $U=X_1/X_2$, can we find $f_{X_1}(x_1)$ and $f_{X_2}(x_2)$?

Problem Statement: Let $X_1$ and $X_2$ be independent random variables. Given the density function, $ f_U(u)$, and the relationship: $$U=X_1/X_2$$ Can we find the density functions for $X_1$ and ...
0
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1answer
285 views

Probability distribution exercise

I'm never certain about my solutions on probability problems (applied), analytic probability and measure is good. I want to work as a TA on a engineering course, so i'm preparing myself doing ...
0
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1answer
79 views

Beta.dist in excel

I've been trying to understand how the Beta.dist function works in MS Excel. The MS Office help says that Beta.dist returns the beta cumulative distribution function. The FALSE of the function gives ...
0
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1answer
67 views

Drawing by replacement and without

I am still studying statistics (almost done with the basics). I read something interesting in a document from Berkeley: "Since $\Bbb E[x] = \mu$, we say that the sample mean is an unbiased estimator ...
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2answers
72 views

Distance of two normal distribution functions

Let $\Phi(u)$ be the standard normal distribution function. We want to consider the following expression $\|\Phi(u/\sigma_l)-\Phi(u)\|_{\infty}$, where $\sigma_l\xrightarrow{l\rightarrow\infty}1$. For ...
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1answer
75 views

Can a Probability distribution be “random”

A quick question. I know many models probability models exist (normal, uniform, etc.). Wikipedia has a nice article listing them all (almost). My question is simple, you have a population defined by a ...
1
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0answers
43 views

Division of Dependent Random Variables

Let $X_1$ and $X_2$ be dependent random variables. Find the density function for: $$U=X_1/X_2$$ Can I use the transformation method to find the conditional density and then use method of ...
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2answers
3k views

Probability of sampling with and without replacement

In sampling without replacement the probability of any fixed element in the population to be included in a random sample of size $r$ is $\frac{r}{n}$. In sampling with replacement the corresponding ...
2
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1answer
61 views

How to finish some complex integration

How to finish some integration as following below: $$\int_x^{\infty} \frac{\mathrm \beta^{\alpha+\gamma} X^{\alpha-1}(y-x)^{\gamma-1}\exp^{-\beta y}}{\Gamma(\alpha) \Gamma(\gamma)}dy\;$$ and ...
0
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1answer
120 views

Normalizing a 3D angle distribution

I'm having trouble finding the proper keywords to search for this type of treatment so I apologize in advance if this is quite obvious. I have a collection of lines in 3D space approximately centered ...
0
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1answer
2k views

Cumulative Distribution Function of Logistic Distribution.

pdf of logistic distribution is : $$p_X(x)=\frac{\pi}{\sigma\sqrt 3}\frac{\exp[\frac{-\pi(x-\mu)}{\sigma\sqrt3}]}{(1+\exp[\frac{-\pi(x-\mu)}{\sigma\sqrt3}])^2};\quad-\infty<x<\infty$$ I have ...
0
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2answers
125 views

Normal distribution probability problem.

There are lots of salmon in a pond and their length (in centimeters) obeys normal distribution $N(70, 5.4^2)$. You and your friend go fishing and decide to continue fishing until both of you catch at ...
0
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1answer
219 views

Convolution of two dimensional gaussian functions

I want to calculate the sum of two probability density functions. I know that it is: $P_{U+V} (x)= (P_{U} * P_{V})(x)$ If $P_{U}$ and $P_{V}$ are gaussian functions in one dimension, i.e. $P_{U}(x) ...
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1answer
2k views

Difference between at least and less than binomial probability

Got a two part question given to me (I used binomial Distribution to solve) If the probability that an individual moves outside of his or her country of residence in a given year is $0.12$, what ...
3
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1answer
48 views

Independent representation of correlated $N(0,1)$ variables

Assume that $X_1$ and $X_2$ are correlated $N(0,1)$ variables. Now we can write \begin{align*} (X_1,X_2)^{T}=(\tilde{X_1},\gamma \tilde{X_1}+\sqrt{1-\gamma^2}\tilde{X_2})^{T} \end{align*}where ...
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1answer
33 views

Simplifying a hpergeometric distribution

So I have a hypergeometric distribution where there are 50 firms and an inspector is going to visit 10 of them to check for a violation. If 15 firms actually are in violation, then the pmf of this ...
0
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1answer
282 views

Expectation problem in Absorbing Markov Chain(exercise on Grinstead and Snell 11.2 18 )

Hi I encountered this problem. It took me quite long but I could not solve it. The problem is as follows: Assume that a student going to a recently established school in a university has, each year, ...
0
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1answer
102 views

How to find the distribution of x-y considering inverse guassian

Let X and Y both be distributed Inverse Guassian which are independent, what is the distribution of Z=X−Y? is there any closed form for distribution of Z?!
0
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1answer
31 views

How does one go about finding a distribution for this property of the distribution?

I am told I need to find a probability distribution in which this Chebyshev Inequality is fulfilled: $P(|X-\mu|\ge 5\sigma)=.04$. What I tried was just taking a simple distribution where the support ...
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0answers
69 views

What does a partition function tells you?

I have some difficulties to really understand what does a partition function say about your data/observations. For instance, say that we have a price series $P(t)$ on the time interval $[0,T]$ and ...
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0answers
181 views

KL divergence of multinomial distribution

Consider $q(x)$ be a Multinomial distribution over $\{1, \ldots, k\}$ with parameters $\{\theta_1,\ldots, \theta_k\}$. And p(x) over $\{1,\ldots, k\}$ with distribution $p(x)=\frac{1}{k}$. Then what ...