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

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17 views

Expected minimum size of multinomial distribution that includes at least one value of each class [duplicate]

Suppose I have a vector and I include a new item on it at every time step. This item has one of $k$ different classes, each class has probability $\frac{1}{k}$ of being selected. Suppose now I stop ...
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0answers
51 views

Does this probability statement have a closed form? (Extreme value distribution)

Problem Statement: Does this probability statement have a closed form solution? $\mathbb{P}\left(\min\left\{ w,p\right\} >c\right)=\mathbb{P}\left(\min\left\{ \left(\frac{a+\epsilon_{1}-\...
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1answer
28 views

Distribution function of uniform distribution

I don't know why the distribution of this question is that when x is in between 0 and theta. In solution.. is that right? I searched the distribution of uniform distribution. But it is alike with that ...
2
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0answers
12 views

Is it possible to conveniently express the probability density of a vector by its Fourier transform?

Let $x$ and $\hat{x}$ be a random vector and its Fourier transform, respectively. For any practical purpose we can assume it's a finite vector and the Fourier transform is given by the DFT, i.e. there ...
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0answers
17 views

Finding the CDF of a joint distributiion

Let $X$ ~Bernoulli(1/3) and $Y = 4X -2$ Find $F(x, y) = P(X \le x, Y \le y)$ I found it to be: $ F(x, y) = \begin{cases} 0 & min\{x, (y+2)/4\} < 0 \\ 2/3 & 0 \le min\{x, ...
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1answer
31 views

Equilibrium distribution exponentially fast

I need to prove that for an aperiodic, irreducible Markov Chain $X_n$ with stationary distribution $\pi$ holds that $P_x[X_n=j]\to\pi(j)$ exponentially fast. I found some proof of that statement but ...
5
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2answers
65 views

Average shortest distance between a circle and a random point lying in it

What is the average shortest distance between the circle $(x-a)^2+(y-b)^2=r^2$ and a random point lying in it? This question is just idle curiosity. Basically, it's the same as finding the ...
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0answers
92 views

Probability of losing lotteries needed

A person earns $x_i$ amount of money every month where $x_i$ is an exponential random variable with parameter $\lambda_1$. The amount $x_i(1-p)y$, here $0 \leq p\leq 1$ and $y$ is exponentially ...
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1answer
49 views

Simulating Probability Distributions

We have the following cumulative distribution function: $$ F(x) = \begin{cases} 0 & x < 0 \\ x^2/9 & 0 \le x \le 3 \\ 1 & x > 3 \end{cases} $$ To find $X$ in terms of $U \sim ...
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0answers
30 views

Random variables have equal moments

I have to show that if two random variables $X,Y$ are bounded and for all natural $n$ $EX^n=EY^n$ then they have the same distribution and the assumption that they are bounded is necessary.I've ...
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0answers
20 views

Showing independence of two independent identically distributed random variables.

The title might be confusing, here is the task: $X$ and $Y$ are two independent and identically distributed random variables with $\mathbb{P}(X=-1)=\mathbb{P}(X=1)=0.5$ and $Z=X\cdot Y$. Show that $...
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1answer
32 views

What is $cov(Y_1,Y_2)$ if $Y_1 = \prod_{i=1}^{n}X_i$,$Y_2 = \prod_{i=2}^{n+1}X_i$ , and $X_{1..n+1}$ are $n+1$ bernoulli-distributed random variables?

$X_1,X_2,...,X_{n+1}$ are $n+1$ random variables that follow a Bernoulli distribution with parameter $p$. We define $Y_1 = \prod_{i=1}^{n}X_i$ and $Y_2 = \prod_{i=2}^{n+1}X_i$. How can I find out ...
2
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1answer
37 views

Name of the distribution with density $P(x) e^{-x/\theta}$, where $P$ is a polynomial with positive coefficients.

The Gamma distribution of shape $k$ and scale $\theta$ has density $\frac1{\Gamma(k)\theta(k)} x^{k-1} e^{-x/\theta}$. Consider the more general distribution with density (up to a normalizing constant)...
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1answer
20 views

Convergence in distribution: constant multiplication

If I have $$X_n/\sigma$$ converging in distribution to $\mathcal{N}(0,\sigma^2)$, does this mean I can just multiply through with $\sigma$ and obtain that $X_n$ converges to a standard normal? I ...
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0answers
29 views

Family of distributions closed under mixture and scaling?

I am looking for a family of distributions that is closed under mixture and scaling (by scaling, I mean stretching the CDF along the horizontal axis). I have thought a lot about this, and have found ...
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0answers
29 views

Given the joint pdf of $X$ and $Y$, find the joint pdf of $W = X+Y$ and $T = X-3Y$.

Let the two-dimensional random variable $(X, Y)$ be whose joint probability function is: $f(x,y) = 1/4$, for $0\le x\le2$ and $0\leq y\leq 2$ a) Calculate the joint probability density ...
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1answer
27 views

Probability Uniform Distribution 4 [closed]

Suppose that X has a uniform distribution on the interval $(0, a)$, where $a > 0$. Find $\text{Pr}(X > X^2)$. I feel that if $a\le1$ then the above probability will be 1(not sure though). But ...
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0answers
24 views

solving for lagrange multiplier in function of distribution

This is the first time I have encountered a functional derivative. My background is not in Physics so this functional derivative concept is quite perplexing to me. I have a function of distribution ...
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1answer
31 views

Calculating PDF from Autocorrelation

I have a statement like this; A zero mean Gaussian random process $X(t)$ is wide sense stationary with the auto-correlation function $R_x(\tau) = 4e^{-2|\tau|}$ And I want to find the ...
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1answer
17 views

Dual normal distribution

Let random variable X has dual normal distribution with mean vector $\mu$ and covariance matrix $\sum$ $$X= \left ( \begin{array}{ccc} x_1 \\ x_2 \end{array} \right )$$ $$\mu= \left ( \...
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0answers
30 views

Probability // Dice problem [duplicate]

I couldn't figure out the solution of it. Assume we are tossing a fair dice 3 times. Describe the probability space related to this experiment and calculate the probability that we have tossed ...
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0answers
22 views

Simple question to check property of Normal Distributions.

So I was preparing for an entrance test and the study material provided by the institution has a lot of mistakes .Please forgive me if this seems off topic. I just know high school maths and I am ...
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1answer
33 views

Let $X\sim\text{Geometric}(1/5)$. Compute $P(X^2 \leq 15)$.

Let $X\sim\text{Geometric}(1/5)$. Compute $P(X^2 \leq 15)$. I came up with $P(X\leq 3)=.5904$, but I'm not sure of my methodology. Applying the square root to each side of the equation gives $P(-3....
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2answers
47 views

Work on transformation of variables

I have $X,Y \sim Uniform(0,1).$ I want to find the PDF for $X^Y$. I imagine that I should start with the CDF $$F_{X^Y}(x) = P(X^Y \leq x) = P(X^y \leq x | Y = y)P(Y = y),$$ But I seem to be having ...
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3answers
41 views

Probability Question for Random Variable $R = \sqrt{X^2 + Y^2}$

Problem: Let $(X, Y)$ be uniformly distributed on the unit disk $\{ (x,y) : x^2 + y^2 \le 1\}$. Let $R = \sqrt{X^2 + Y^2}$. Find the CDF and PDF of $R$. Attempted Solution: First note that $r \in R = ...
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1answer
102 views

finding Expected Value for a system with N events all having exponential distribution

We have a system in which events happen one after each other. The time interval between each two events shown by random variable $t_i$. So, the time interval between the first and the second events is ...
1
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2answers
38 views

Sum of Product of Normal Random Variables

Consider a collection of $n$ i.i.d.normal random variables $X_i \sim \mathcal{N}(\mu, \sigma^2)$, $i=1,\ldots,n$. I'm trying to compute the distribution of $$ \sum_{i=1}^n \sum_{j=i+1}^n X_i X_j. $$ ...
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1answer
38 views

Joint pdf of X and Y with absolute value

Question. Joint probability function of continuous probability X, Y is here : $f_{X,Y}(x,y) = k(|x|-|y|) \ \ \ \ \ \ \ \ \ \ (-1< y< x< 2)$ Then what is k? I mean how can I differentiate ...
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0answers
38 views

How to construct uniform probability distribution over d-dimensional convex hull?

I'd like to sample random variables and/or calculate likelihood of given random variables that have uniform probability distribution over a d-dimensional convex hull. The convex hull is given as a ...
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0answers
88 views

What is the value of the following probability? [closed]

Let $f_{WE}(t,\alpha,\lambda)=\alpha \lambda t^{\alpha-1}\exp\{-\lambda t^\alpha\},~ t>0.$ The joint PDF of $(X_1, X_2)$ is given as follows $$ f(x_1,x_2)=\left\lbrace \begin{array}{ll}‎ f_1(x_1,...
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2answers
29 views

Probability Distribution of Random Variable

If our random variable only has six equal possible outcomes, will any probability distribution resulting in mapping to real numbers consist of only six real numbers each with probability $\frac{1}{6}$ ...
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0answers
24 views

HMM limiting distribution

Consider a hidden markov model (HMM) with two hidden states $A$ and $B$ and emission support $1$ and $2$ fitted with initial state distribution $$\lambda = [\begin{array}{cc} .7&.3\end{array}]$$ ...
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0answers
54 views

Distribution of the ratio of two generalized gamma distribution with same location parameters?

X $\sim$ GG(p,d,$\theta_{1}$,$\mu$) where p is power, d is shape, $\theta_1$ is scale and $\mu$ is location parameter. Also Consider Y $\sim$ GG(p,d,$\theta_{2}$,$\mu$) where p is power, d is shape, $\...
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1answer
65 views

If $X$ and $Y$ are identically distributed then $(X,Y)$ and $(Y,X)$ are identically distributed?

If $X$ and $Y$ are identically distributed then $(X,Y)$ and $(Y,X)$ are identically distributed? I think the answer is no but I couldn't find a counterexample. And one more question. If $(X,Y)$ and ...
2
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1answer
14 views

Graph the describes the peaks of a Poisson Distribution

Probably not a very smart question - barely know any of the more interesting parts of probability theory - but I noticed that the peaks of Poisson curves form what looks like a kind of logarithmic ...
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0answers
26 views

Probabalistic modeling of graph topology / network structure

I'll just let you know right now that I will be using very informal language here, so if you have other questions about technicalities that need to be specified please let me know. Let's say we have ...
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2answers
27 views

Sigma Sum Properties

My Probability and Statistics teacher wrote the following: http://imgur.com/z0rgylh How can I prove such step? Thank you very much!
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1answer
46 views

Is there a distribution which looks like this?

I was playing around with some data. And I got a distribution which looks like the following. Does this resemble any of the known distributions?
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0answers
35 views

Characteristic function of a lattice distributed random variable

Let $X$ be a random variable. $X$ is called lattice distributed if there exist real numbers $a, b$ such that $P(X \in a +b\mathbb{Z})=1$. Show that $X$ is lattice distributed if there exists $v\neq 0$ ...
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0answers
43 views

distribution of random variable nad expectation value

We throw a die as many times as we thrown $k$ tails and $k$ heads (it can't be one after another). Let $X$ be a number of throwing. Set the distribution of $X$ and $\mathbb{E}X$ So, I've started from ...
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1answer
23 views

Convolution of the cumulative normal distribution and the uniform distribution [closed]

What is the resulting function of convolving the cumulative normal distribution and the continuous uniform distribution?
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161 views

Can we model this set of experiments as an stochastic process and estimate the sample size?

I have an image with the size 5575x9440 and I'm implementing a modified version of the algorithm used in this paper on it, but because the code performance is low ...
2
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1answer
13 views

Finding the mean given the PDF of the Pareto Distribution

I'm trying to determine the general PDF and Mean for the Pareto distribution description of the size of TCP packets, given that distribution's CDF: $$ F(x) = \begin{cases} 1-\left(\frac{k}{x}\right)^...
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0answers
12 views

Characteristic function of triangular distribution over $[0,2]$.

Here it is shown that if $X_1,\dots ,X_n$ are iid random variables, then the characteristic function of $S=\sum_{i=1}^nX_i$ is the product of the respective characteristic functions of the $X_i$. ...
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1answer
21 views

Find constants so that a random variable has a chi-squared distribution

Suppose $X$ ~ $N(3,5)$ and $Y$ ~ $N(-7,2)$ be independent. Find constants to satisfy: $\quad C_1(X+C_2)^2$ + $C_3(Y+C_4)^2$ ~ $\chi^2(C_5)$ I started with: $\quad Z_1=\sqrt{C_1}X+\sqrt{C_1}C_2$ ~...
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1answer
26 views

calculate the variance and expectation

So we have $50$ white balls. We draw balls one after another with returning and if it is white ball we repaint it into red one. Let $X$ be the number of red balls after $20$ drawing. Calculate $\...
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0answers
31 views

Probability - binary array through a comm. channel

An information through a communication channel is transmitted as a binary array - an array of bits (1s and 0s). But the bits can get corrupted, 1 can turn into a 0 and 0 can turn into a 1 with a ...
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1answer
29 views

Given $f(x,y) = 1$, $0<x,y<1$, let $U = X+Y$. Find $f_U(u)$.

Would anyone be able to explain what they did in the second line, especially how they got $0<u<1$, and $1<u<2$
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1answer
35 views

Discrete Uniform Probability: isn't my textbook just wrong?

My textbook is showing me examples of discrete probability distributions, one of them is in the picture: I learned in Calculus that the summation of the series $1/n$ where $n\to \infty$ is ...
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1answer
27 views

What type of distribution is this?

$$p(x) = \frac{1 }{\theta} . \frac{(ln\theta)^x}{x!} $$ I have no idea what type of distribution is this? And what will be its first and second moment?