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

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15 views

Let X and Y be continuous random variables with joint PDF of the form $f(x,y) = c(x+y)$. Find the joint CDF

Let $X$ and $Y$ be continuous random variables with joint pdf of the form $f(x, y) = c(x+y)$ $0 < x < y < 2$ and zero otherwise. a. Find c so that f(x, y) is a joint pdf. I answered this ...
2
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2answers
28 views

Are there two different notions of “conditional probability”?

This question comes from reading the discussion here. (1) If one is given a "probability measure" $P : F \rightarrow [0,1]$ mapping a Borel $\sigma$-algebra $F$ to $[0,1]$ then for two ``random ...
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2answers
29 views

Probability that one chi-squared random variable is less than other chi-squared random variable

I have two random variable $X=\mathcal N(\mu,\sigma^2)$ and $Y=\mathcal N(0,\sigma^2) $ independent to each other. Now, $Z=X^2$ and $W=Y^2$, are chi-square random variable with first degree of ...
2
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0answers
11 views

The uniqueness of solution of an equation that involves CDFs

I have two monotone CDFs $F(x)$ and $G(x)$. The functions are symmetric in a sense that $F(x)=1-G(1-x)$, $f(x)=g(1-x)$. I am trying to show that equation $xF(2x)+(1-x)G(2x)=1/n$, $n\geq2$ has a unique ...
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1answer
10 views

Cumulant-Legendre

I have a short question: So suppose $b=\text{ess sup} X<\infty$, where $X$ is a random variable on $\mathbb{R}$. Now take $\Lambda (u)=\ln \mathbb{E}[e^{uX}]$, the cumulant, and ...
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1answer
15 views

Let X and Z form a random sample from a poisson dist.If Y=min( X,Z), what is P(Y=1)??

Let X and Z form a random sample of poisson distribution and define Y=min( X and Z) What is P(Y=1)?? I think Y is minimum of two. If X=1, then Z can be any number except 0 If Z=1, then X can be ...
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1answer
27 views

The size of a fish in a lake follows a normal distribution

I have a homework question that I wasn't positive about. This is the first probability course I have taken and the class is only taught using excel so I apologize for the lack of formulas in my ...
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1answer
20 views

Lifetime of light bulbs is modeled as a Poisson Process - using excel

I have a homework question that I can't seem to figure out. Any help is appreciated! The lifetime of light bulbs (in days) is modeled as a Poisson Process with expected lifetime of beta = 200 days. A ...
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1answer
17 views

Using PMF and CDF to calculate probability

Given the following CDF what is $$P(T > 3)$$ and according to my answer key it's 1-1/2 = 1/2. Can someone explain to me why it is 1-F(3), and would subtracting F(3) be subtracting 4 as well? ...
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1answer
24 views

Comparing Percentiles of 2 Samples Drawn from the Same Distribution

Suppose I have two sets of numbers: $A=\{a_1,a_2,...a_{N_1}\}$ and $B=\{b_1,b_2,...b_{N_2}\}$ with $N_1<N_2$. WLOG assume that $a_i<a_j$ for all $i<j$ and similarly for $b_i$ and $b_j$. ...
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1answer
33 views

Distribution of minimum absolute value

Consider $K$ independent Laplace variables $X_k, k=1,\ldots,K$, with mean 0 and scale $\lambda$ (so that their PDF is $f(x)=\frac{1}{2\lambda}e^{-\frac{|x|}{\lambda}}$. Let $Y$ be the variable taking ...
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1answer
21 views

Given Nd6, what is the probability that the two highest are minimum 4?

So, my statistics knowledge is rather poor, so I would welcome a formula explanation to the question: given Nd6 (6-sided dice) what is the probability that the two highest numbers are at least a 4? ...
2
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2answers
43 views

Does the distribution of a process on $\mathbb{R}^{[0,\infty)}$ uniquely define it?

Question: Can I have two different stochastic processes $(A_t)_{t \in [0, \infty)}$, $(B_t)_{t \in [0, \infty)}$ having the same distribution on $\mathbb{R}^{[0, \infty)}$ differ in some ways? ...
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2answers
16 views

Probability of X given the sum

I am given that $X \sim P(\lambda)$, $Y \sim P(\gamma)$, and told to calculate the distribution of $P(X | X+Y = n)$ I proceed as follows $$ \begin{equation} \begin{split} P(X=i|X+Y=n) &= ...
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1answer
23 views

Exponential distribution question!

Suppose that the time between calls from your best friend has an exponential distribution with a mean time of $3$ days. (a) If you just received a call from her, what is the probability that you will ...
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1answer
24 views

How to show that $E(X^k)=npE((Y + 1)^{k-1})$ where $X\sim\mathrm{Bin}(n,p)$ and $Y \sim \mathrm{Bin}(n-1,p)$.

Show that $$E(X^k)=npE((Y + 1)^{k-1})$$ where $X\sim\mathrm{Bin}(n,p)$ and $Y \sim\mathrm{Bin}(n-1,p)$. I am looking for suggestions on where to start? Or any resources someone may have. I am not ...
2
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0answers
24 views

convolution of two probability density functions

Please no one call me dumb - I am not a mathematician and haven't done proper math for the last ten years. But I have a problem at work where I need to perform a convolution of two probability density ...
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0answers
18 views

distance distribution in Poisson point process

Consider a homogeneous Poisson point process in 2D space with density $\lambda$ per unit area. Let $\mathcal{B}(o,R)$ denote a disk centered at origin with radius $R$. Let $n$ be the number of points ...
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1answer
19 views

Finding this Probability Density Function

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1)$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ with $\lambda > 0$ Then calculate $P(Y>t+s|Y>t)$ for ...
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0answers
27 views

Why does symmetry happen in reset-based random walks?

I am studying the basic concepts about random walks / brownian motion, and based on the idea of a Möbius-based walk in Wolfram's website, I wanted to try my own version of it in Python to compare it ...
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0answers
21 views

Number of same degree vertex pairs between two random graphs

I am considering the random graphs generated by the Erdős-Rényi model for this question. Random Graphs as Models of Networks by Newman is a reference on this topic. A random graph $\Gamma_{n,p}$ has ...
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2answers
24 views

Essential supremum via cumulant

Let $p(t)=\log \mathbb{E}[\exp (tX)]$ for $X$ real valued random variable. Now it holds (assuming that $p$ is smooth and finite on $\mathbb{R}$) that $p'(\infty)=\text{ess}\sup X$. How can I prove ...
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1answer
19 views

If $E(|X|)<\infty$, how do we show that it can be expressed as below

$F(x)$ is the distribution function of $\mathbb X$, and $f(x)$ is the derivation of $F(x)$, Prove that $\int_{0}^{\infty}(1-F(x))dx-\int_{-\infty}^{0}F(x)=E(X)$. Note that ...
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2answers
54 views

Why birthday distribution is not uniform. [on hold]

I was reading about birthday problem and I found a statement that real-life birthday distributions are not uniform since not all dates are equally likely (last line ...
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1answer
30 views

Statistics - Exponential distribution

There are $n$ machines. Each has durability given by exponential distribution with $EX = 10$. If a dead machine is replaced with new one immediately, find minimal $n$ so we can say with $P = 0.99$ ...
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0answers
22 views

Probability density function of an uniformly distributed random variable [on hold]

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1).$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ for $\lambda>0$ And calculate $P(Y>t+s|Y>t)$ for ...
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0answers
17 views

Find marginal distribution (Integral Solution)

I have derived bivariate exponential distribution in term of polar coordinate system. Now I need to derive marginal distribution of $f(\theta)$ from joint $f(r,\theta)$ for this we have to eliminate ...
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2answers
115 views

Game of probability

n a game, played between $2$ players there is a circular field and one of the players is blindfolded, who stands in the center of the field. The other player stands at a fixed point on the ...
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0answers
26 views

How do simulate a skewed normal distribution using dice [on hold]

I saw this which suggests 5 dice is enough to simulate a normal distribution. http://www.johndcook.com/blog/2009/02/10/rolling-dice-for-normal-samples/ How can you simulate a skewed normal ...
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2answers
31 views

Find the density of their average

If $f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$ find the density of their average $\frac{X+Y+Z}{3}$ I'm a little lost on how to solve this exercise, ...
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0answers
14 views

Family of parameterized distribution functions with separable inverse

I am looking for parameterized families of distribution functions that have a separable inverse. For example, the inverse of the exponential distribution is $H^{-1}(p;\gamma)=\frac{1}{\gamma}ln(1-p)$ ...
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1answer
40 views

Statistics - normal distribution problem

Two random variables $X$ and $Y$ are i.i.d. normal$(\mu, \sigma^2)$. If $P(X > 3) = 0.8413$ find $P((X+Y)/2 > 3)$. The result must be exact number, so normal distribution parameters are ...
2
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1answer
28 views

Inequality for the derivative of a density of a random variable convolved with a normal r.v.

I have a question about the following proof. The statement is: Let $X$ be a random variable and $Z_\tau \sim N(0,\tau)$ be an independent random variable. Then $Y_\tau := X + Z_\tau$ has a ...
2
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1answer
33 views

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are ...
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1answer
49 views

Is this a misuse of the term “probability space”?

Let me first state the definitions as I am using them. Do correct me if I am wrong here! A "probability space" is a triple $(\Omega, F \subseteq 2^{\Omega}, \mu : F \rightarrow [0,1])$. The ...
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0answers
17 views

(Almost) Gaussian distribution

I need to find a distribution for random variable $\boldsymbol \epsilon$ goverened by parameter $\alpha >0$, such that: for any given $\boldsymbol \pi \in \mathbb [0, 1]^M: (\boldsymbol ...
3
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0answers
23 views

Random matrices, eigenvalue distribution.

I just investigated randn(1024) + 1i*randn(1024), a 1024x1024 complex valued matrix with elements both real part and imaginary part drawn from $\mathcal{N}(\mu = 0, \sigma = 1)$. I was a bit surprised ...
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2answers
29 views

Is my probability reasoning here correct?

Sheldon Ross theoretical exercice A jar contains $n$ chips. Suppose that a boy successively draws chips from the jar, each time replacing the one drawn before drawing another. The process continues ...
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0answers
26 views

Find the Lipschitz constant of a multi-variate Gaussian density function

I would like to find the Lipschitz constant of a multi-variate Gaussian density function: $$f_{\mathbf x}(x_1,\ldots,x_k) = \frac{1}{\sqrt{(2\pi)^{k}|\boldsymbol\Sigma|}} ...
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1answer
13 views

Density of a distribution function at upper bound [on hold]

Consider a strictly increasing continuously differentiable distribution F with support on $[a,b]$. Let $f$ be the pdf of $F$. What can we say about $f(b)$? Under what conditions is $f(b)>0$? ...
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0answers
23 views

A Question about the Kurtosis

Problem: Show that if a binomial distribution with $n = 100$ is symmetric, its coefficient of kurtosis is 2.9. Answer: First, I am interpreting the term symmetric to mean that $p = q = \frac{1}{2}$. ...
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2answers
25 views

The distribution of the product of Gaussian variable and Rademacher variable.

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same ...
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1answer
15 views

example of convergence in distribution but not in probability

While I was looking for an example of a sequence of random variables which converges in distribution, but doesn't converge in probability, I have read that it should be enough to consider a sequence ...
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0answers
6 views

generate binomial binary data

How does one generate correlated binomial data when one is given marginal probabilities of each and also the correlation coefficient. The following code in SAS for example works best when we want ...
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1answer
28 views

Prove that the increments of the Brownian motion are normally distributed

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
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0answers
14 views

Optimal decision for sampling a distribution.

I was wondering which probability distribution is best sampled with $\pm\alpha^n, n\in\{1,2,\cdots\}$ for various values of alpha. Sampling means to pick the one which is closest, store the sign and ...
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1answer
29 views

Failed to understand basic approach of mathematical modeling in research ideas [on hold]

I am very new to mathematical modeling. I study different probabilistic approaches but fail to understand how it can be implemented over any research activity. Is there any set of rules to apply for ...
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0answers
25 views

Closeness in distribution implies closeness in statistics?

I am aware that convergence in distribution does not necessarily imply convergence in the mean. I browsed through some examples of statistical distances here ...
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0answers
16 views

Copula vs Exprimental Copula

I have read some texts about finding/approximating copulas for a given sample based on known (famous) copulas. My question is: when we have the experimental CDF of (X, Y), why we should try to find a ...
1
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1answer
27 views

How to interpret a p-value that's significant from Fisher's Exact test

Given a binomial distribution with p=.03, n=902, the $.025$ and $.975$ quantiles are $17$ and $38$ respectively. I interpret ...