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Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

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21 views

What is the probability distribution of find n points inside a radius in space?

Given a space with a uniform density $⍴$ of points, what is the probability distribution of finding $n$ points inside a radius $r$.
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11 views

From table of discrete joint distribution of $X$ and $Y$ to read the distribution of $X$ and $Y$ and $\mathbb{E}(Y|X)$ — Durrett 1.4.2

This is Durrett $3^{rd}$ Example 1.4.2, I understand most of it but I am stuck in the end. This example is designed to show that $\mathbb{E}(XY)=\mathbb{E}(X)\mathbb{E}(Y)$ does not necessarily imply $...
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1answer
26 views

Deriving the asymptotic distribution

Let $x_1, ..., x_n$ and $y_1, ..., y_n$ be two independent random samples from $X$ and $Y$. We have $µ_X = E (X ) > 0, µ_Y = E (Y ) > 0$ and $σ^2_X = Var (X )$ and $σ^2_Y = Var (Y )$. Derive ...
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25 views

A proof for two random variables in probability

I want to prove the following theorem: Let the random variables $X$ and $Y$ have the distributions $f$ and $g$ respectively where $f$ and $g$ are continuous in $\mathbb{R}$. $X$ and $Y$ have the same ...
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9 views

(Kind Of) Maximising the Variance of a Hypergeometric Distribution

I am playing around with a hypergeometric distribution. Consider an urn with $N$ balls with $R$ red balls and $B$ blue balls. Where $1\leq n\leq N$ is a sample size, and $m\in(0,1]$ a margin of ...
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0answers
15 views

Probabilities of points on n-sphere

I have a process that generates random points on a n-sphere. Given a new point $x$, I want to know how probably the new point comes from the same process or distribution? How should I model the ...
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25 views

Conditional Expectation given n?

At a 24-hour movie theater, customers arrive at a rate of 10 customers per hour. Any given customer will independently buy a ticket for one of the following movie genres with corresponding ...
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21 views

Infinite Sum of Random Variables $X_i$ with $P(X_i=0)=\frac{1}{2}$ and $P(X_i=\frac{1}{i^m})=\frac{1}{2}$ [on hold]

Given $m>1$, define the random variables $X_i$ with $P(X_i=0)=\frac{1}{2}$ and $P(X_i=\frac{1}{i^m})=\frac{1}{2}$. Now let $X=\sum_{i=1}^{\infty}X_i$. I'd like to ask what is the density of $X$?
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1answer
22 views

Showing that exp(-|x|^p) is not a characteristic function [on hold]

I need to show that exp(-|x|^p) is not a characteristic function of a non negative pdf for p>2. I am a bit lost as to how to approach this problem. Thank in advance
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8 views

Auto regressive Process and CLT for dependendent variable

I want to prove the following two things for auto-regressive process $Cov(X_1, X_j)=\beta^(j-1)\sigma^2$and $Var[\sqrt{n})(\bar{X}-\theta)\rightarrow \sigma^2\frac{1+\beta}{1-\beta}$
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1answer
23 views

Distributions of Markov chains

What's the difference between stationary distribution and limiting distribution of a finite state markov chain? Do stationary distributions always exist and limiting distributions may not necessarily ...
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2answers
22 views

Fair coin and weighted coin

I have a fair coin and a weighted coin which lands heads 75% of the time. I pick a coin at random and flip it 5 times and get heads 4 times and tails once. What is the probability that I picked the ...
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23 views

Find f(y|x), x and y are either max or min of two iid random variables. I don't know either x is max or min, but p = Pr(x=max,y=min)=1-Pr(x=min,y=max)

There are two iid random variables $\xi_1,\xi_2 \sim F(\xi)$. I define two new random variables x and y such that: $$with~probabitity~p:~~~~~~~~x=max(\xi_1,\xi_2),~y=min(\xi_1,\xi_2)\\ with~...
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8 views

How do bounds adjust with derived distributions(AKA transformed RV)?

So I have this function $f(x)=2x^{-2}, x>2$ and $0$ otherwise. Calculate pdf of $Y=\sqrt X$ The solution says that the lower bound becomes $\sqrt 2$, is this because $Y=\sqrt X$? How do the ...
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22 views

Find $Y = e^X$ for uniform distribution [on hold]

Let $X$ be a continuous r.v. with uniform distribution with parameters $a = 0$ and $b = 1$. Define $Y = e^X$, find its p.f.d., and check that it integrates to one.
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1answer
26 views

Expected value of a chain of many probability distributions

Let $X$ be a uniform random number in $0..n$. Its expected value is $n/2$. Let $Y$ be another uniform random number in $0..X$. Finally, let $Z$ obey a binomial distribution with $Y$ trials and success ...
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1answer
39 views

How do you prove the following inequality according to Chernoff's bound?

I came across the following problem when I read the book, "Understanding Machine Learning: from theory to algorithms". You can click the link to download the book. The statement is on Page 399. When ...
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15 views

What happens when passing a Uniform Distributed Random variable into some function, what would be the resultant distribution [on hold]

To approximate $\mu = E[h(X)]$ with X ~ U(0,1) and $h(x) = \frac{1}{x^{1/3}} + \frac{x}{10}$ by Monte Carlo method. using simple sampling to approximate $\mu$ by $\hat\mu_n=\frac{1}{n}\sum_{i=1}^n h(...
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1answer
14 views

Uniform distribution over the square region.

Given: $(X,Y)$ be uniformly distributed over the square $R = \{(x; y) : 0 < x < 1; 0 < y < 1 \}$ . Let $U = min(X; Y )$ and $V = max(X;Y)$. My attempt to show that: $U$ and $V$ are not ...
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9 views

Simulate a model including random variable and known dataset

I took simulation as a tool of uncertainty propagation. My model-for-solving looks like this: $Y= X_1 \cdot X_2\cdot X_3\cdot X_4$ I found good data on $X_1(n=1000)$. For $X_2, X_3 and X_4$ I also ...
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1answer
33 views

Expected value of “composite” probability distributions

Let $X$ be a uniform random number in $0..n$. Its expected value is $n/2$. Next, let $Y$ obey a binomial distribution with $X$ trials and success probability $p$. So now we have a distribution where ...
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1answer
27 views

What is the intuitive explanation of the Joint PDF formula?

I know that $$f_{X,Y}(x,y)=\frac{\partial}{\partial x\partial y}F_{X,Y}(x,y)$$ I am not exactly sure of the intuitive description of doing two partial derivatives. In the one-dimensional case, the ...
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1answer
26 views

Few questions of circularly symmetric complex random variables

Asked here as well: https://mathoverflow.net/questions/346148/three-questions-of-circularly-symmetric-complex-random-variables Let $Z: \Omega \to \mathbb{C}$ be a random variable with density $f_Z$. ...
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0answers
18 views

Exponential identities - transformation of Weibull probability distribution functions

These two functions are identical, both are Weibull probability distribution functions. By fixing $\beta$ and $\alpha$, the plot of $y$ interms of $x$ from both functions are identical. How can I ...
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0answers
27 views

Find the pdf $f_{y}(y)$

Given X is a random variable with pdf $f_X(x) = \frac{2x}{81} $ where $x \in (0,9)$. $Y = X^{2} - 8X$. My attempt is to find the pdf for $Y$ $f_Y(y)$. I found that: $f_Y(y) = \frac{8}{81\sqrt{y+16}}$...
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1answer
34 views

Calculate cumulative probability for X~U(0,1)

I'm trying to calculate $$ P\left( |X - \mu_X| \geq k \sigma_X \right) $$ for $X\sim$uniform(0,1). I've calculated that $E[X]=1/2$, Var$[X]=1/12$, know that $$ f_X(x) = 1, 0 \leq x \leq 1 $$ and ...
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1answer
39 views

Expected Value Problem including permutations

Please help! Let F denote some permutation that maps from A to A. Where A={1,2,...2n} is a set. For such a permutation, F, let p denote the number of indices i, belonging to A, such that $F(i) > ...
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31 views

Does $\phi(t)^2 = \phi(2t)$ imply $\phi$ is the characteristic function of a Cauchy distribution?

Suppose that $\phi(t)=E(e^{itX})$ ($t\in\mathbb{R}$) is the characteristic function of a real-valued random variable $X.$ If $\phi(t)^2 = \phi(2t)$ holds for every $t \in\mathbb{R},$ is it necessary ...
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0answers
12 views

Difference between two independent Beta random variables

I have two variables: $$X \sim \mathrm{Beta}(a_1, b_1)$$ and $$Y \sim \mathrm{Beta}(a_2, b_2)$$ where $X$ and $Y$ are independent. I want to find the PDF of the difference between these randomly ...
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1answer
41 views

What is the probability distribution of this random variable?

Let $X$ be a random variable $\mathcal N(0,1)$. How can we find the distribution of $$Y= \frac{1}{|X| \sqrt{2 \pi}} e^{\frac{-1}{2 X^2}}$$ What are the available tools to solve this problem, or any ...
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0answers
25 views

Expected value of the $k$'th order statistic given that it's smaller than $\tau$

Let $X_1,\ldots,X_n\sim U[0,1]$ be i.i.d. uniform random variables and let $X_{(k)} $ denote the $k$'th smallest variable. Given some $\tau\in(0,1)$, what is $$\mathbb E[X_{(k)}\mid X_{(k)}\le \tau]?$...
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1answer
14 views

How does multiplication by a constant affect a Gumbel random variable

Suppose $X$ is a Gumbel (Type-1 extreme value) random variable with shape and scale parameters given by ($\mu$, $\sigma$). What is the distribution of $cX$, where $c$ is a constant?
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1answer
22 views

Understanding the Glivenko-Cantelli lemma in relation to strong law of large numbers

Consider the following estimator of the distribution function $F$. Let a sequence of a random variables $X_i$ be i.i.d with distribution function $F$ and our estimator $\hat{F}_n(X) = \frac{1}{n} \...
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1answer
47 views
+200

Uniform prior distribution on log scale

Can anyone please suggest me a distribution whose $log$ transformation is uniform and it should be a well known distribution? I am not sure if it exists. I know that if we consider a uniform ...
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2answers
30 views

Calculate probability $P(\text{min}\{X_1, \ldots , X_k\} = X_j )$

Suppose the exponential random variables $X_1, \ldots X_k$ are independent, each with parameter $\lambda_i$, respectively. Calculate $P(\text{min}\{X_1, \ldots , X_k\} = X_j )$, for $j \leq k$. I ...
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2answers
24 views

Formula probability of two random variables with density function

I am reading through a proof in Brzezniak's Basic Stochastic Processes on the Poisson distribution. If $\xi$ and $\eta$ are independent random variables and $f_{\eta}$ is the density of $\eta$ then ...
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1answer
21 views

exponential distribution cups breaking

let $X$ r.v. denote the time needed for the event of a cup breaking with exp$(\frac{1}{24})$ distribution, in other words a cup breaks every 24 months(expected value). If I want to know what is the ...
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2answers
22 views

P(X<Y) given joint density function

I am given the joint density function $f(x,y)$ for random variables X,Y with $0<x<1$, $0<y<2$ I am interested in $P(X<Y)$ My first instinct was to do the following: $$P(X<Y) = \...
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0answers
24 views

Estimating the mean using Chebyshev's inequality [on hold]

I found the following problem in my homework and I was hoping you guys could help me out: Consider a random circle where the radius is given by a random variable uniformly distributed along the ...
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1answer
42 views

Are these two R.V. independent?

Let us consider the following R.V. $S_1=S_0e^X_{1}$ and $S_2=S_1e^X_{2}$, where $X_1 \sim N(\mu_{1},\sigma_{1})$ and $X_2 \sim N(\mu_{2},\sigma_{2})$ are independen.$S_0$ is a constant. I know that ...
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1answer
56 views

What is the distribution of the product of these two R.V?

Let $X$ be a random variable with gaussian distribution $N$ ($\mu$, $\sigma^2$). Suppose $Z$ is a Bernoulli $B(p)$, independent of $X$. Find the distribution of $Y = (2Z − 1)X$ Here is my attempt: ...
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2answers
47 views

Density of a point

You choose a random point inside the triangle with vertices $A=(0,0),B=(1,3),C=(2,0)$. Let $(X,Y)$ be the random double variable that indicates the point chosen. Find the density of $(X,Y)$ Ok, we ...
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3answers
55 views

Density function of product

Suppose $X$ and $Y$ are continuous random variables with joint density $$f(x,y)=x+y,\quad 0<x,y<1$$ I am trying to find the density of $XY$. I am having trouble applying the formula $$f_V(v)=...
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0answers
21 views

Probability of rolling a dice within a stripe [on hold]

A group of friends are playing a game that involves rolling a standard 6-faced die on a table. The table is covered with a black-and-white striped tablecloth, and one friend notices that, in ...
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1answer
40 views

Find the joint distribution of (X, Y) where $X = N(\mu, \sigma^2)$ and $Y = I_{\{X > 0\}}$

Suppose that $X \sim N(\mu, \sigma^2)$ and $Y = I_{\{X > 0\}}$. What is the joint distribution of the random vector $(X, Y)$? I have thought two possible solution for this: (i) By the formulation ...
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0answers
22 views

Joining two probability density functions [on hold]

Question I need help with assistance with two probability density functions to make a joint probability density function and to manipulate the function to find X+Y as a density function as well as x/...
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1answer
35 views

Finding the variance of x

The random variables $X$ and $Y$ have the joint probability density function $$ f_{X,Y}(x,y) = \left\{ \begin{array}{ccc} \frac{1}{y}, & 0 < x < y, & 0< y< 1 \\ 0, & \text{...
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1answer
18 views

Exponential Distribution of 2 random variables. [on hold]

I have: 2 random variables: $X_1$ has an exponential distribution with parameter $\gamma_1$ and $X_2$ also has an exponential distribution with parameter $\gamma_2$. Assume that $X_1$ and $X_2$ are ...
0
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0answers
9 views

Marginal probability mass function of bivariate negative binomial

define $$P(X=x,Y=y) = {(x+y+k-1)!\over x!y!(k-1)!}p_1^xp_2^y(1-p_1-p_2)^k$$ the bivariate negative binomial distribution. I am interested in the marginal probability mass function of X. After ...
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0answers
19 views

Pareto Principle demonstration

I was simulating a popular pareto principle demonstration that goes as such: get a bunch of paperclips throw them into a pile link two random chains/paperclips from the pile throw the new chain back ...