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

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9 views

Reconstruct a family of probability distributions having certain generalized hypergeometric moments

Reconstruct and/or otherwise characterize any/or all members of a certain one-parameter ($\alpha =\frac{1}{2}, 1, \frac{3}{2}, 2,\ldots$) family of univariate probability distributions (of ...
0
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0answers
14 views

Calculating the Distribution Function of Chord Length

The question: Choose two (different) points on the circle $S^1 \subset \mathbb{R}^2$ at random (with uniform distribution), and connect them with a straight line. Define a suitable probability space ...
1
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1answer
22 views

How to define $E(X)$ when X is a random variable from the sample space to an infinite-dimensional topological vector space?

Let $V$ be an infinite-dimensional vector space with a topology. For simplicity, we can assume $V$ is a Banach space. Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $X:\Omega \to V$ be a ...
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0answers
23 views

Ways to sample a complicated PDF on an hemisphere

I want to generate samples on the upper real unit hemisphere with the following PDF (it's not really a PDF because I can't guarantee that it integrates $1$) $$\frac{\sum_{i=0}^{n}c_i(\text{ ...
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1answer
22 views

Using Moment Generating Function to prove Z is standard normal

Suppose $X_1,...,X_n$ is a random sample from a normal distribution with an unknown mean $\mu$ ,known standard deviation $\sigma$ and sample population $\bar{X}$. Show (using moment generating ...
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1answer
11 views

How do you find the PDF when you are given the new variable wrp to a known random variable?

My question is rather simple but here's a specific example I'd like to work with. The pareto distribution is given by the PDF $f(y:\theta)=\theta y^{-\theta-1}$ and $y_i$ are distributed with this ...
2
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2answers
20 views

Linear Transformation of Poisson Point Process

Suppose we have a random variable that follows a Poisson Point Process: $ X \sim poisson(\lambda t) $ and a function $f(x) = ax + b $ where $a,b \in \mathbb{R}$. What is the pdf of $Y = aX + b$? I ...
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0answers
17 views

Normal Distribution and Construction of Independent Variable [on hold]

Suppose $X$ and $Y$ are bivariate normal such that $E(X) = E(Y) = 0$ and $Var(X) = Var(Y) = 1, Cov(X,Y) = 1$ Then how can I find a non constant random variable which is a function of $X$ and $Y$ , ...
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1answer
17 views

Compute the distribution function of the random variable $Y:=-ln(F(X))$

I'm trying to prove this result: If $X$ is a continuous random variable with distribution function $F$, where $F$ is strictly increasing function, then find the distribution function of the random ...
-2
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0answers
15 views

Why is the mean of the $t_1$ distribution undefined? [duplicate]

The pdf of the $t_1$ distribution is symmetric about $0$, just like the pdfs of $t_k$ for $k>1$, so why does it not also have an expected value of 0?
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0answers
42 views

approximating a uniformly distributed random variable

Suppose that $U$ is a uniformly distributed (continuous) random variable on $[0,1]$. Let's say that I am interested in finding 3 discrete points $u_1,u_2,u_3$ which approximate $U$ in some sense. My ...
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0answers
53 views

Affine function of cdf [closed]

Suppose a random variable X has cdf FX(·). Express the cdf of the following random variables: X + b aX + b |X| max(X,0) Could someone show me how to use the given random variable X and its ...
0
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1answer
34 views

Rolling a die that follows a poisson distribution and computing it's mean

Roll a fair, 4-sided die N times where N is a Poisson random variable with parameter λ>0, let X be the number of 3's rolled in this experiment. Find E(x) What I have figured out is that E(X) seems to ...
0
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0answers
17 views

Is there a theorem or law stating that the expected value of a symmetrical distribution across $f(X)$ will equal $f(\bar{x})$ iff $f'=c$ across X?

I have noticed that the expected value of a symmetrical distribution across $f(X)$ will equal $f(\bar{X})$ only if $f'$ is constant across X. For example, consider a uniform distribution X ranging ...
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0answers
28 views

Probability//Uniform Distribution [closed]

It is believed that the time $X$ (min) for a lab assistant to prepare the computer for a certain experiment has a uniform distribution with $A = 25$ and $B = 35$. How can I determine the density of ...
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0answers
32 views

Derivation of pdf from the function of random variables [closed]

Let $A_{i}$ and $B_{i}$ ($i=1,...,K$) be the random variables of which pdf/cdf are known to us. And, there is a function of random variables, $C=\max({A_{1}+B_{1}, A_{2}+B_{2}, ..., A_{K}+B_{K}})$. ...
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2answers
31 views

Finding large deviation bound for binomial distribution

$S \sim Binomial(n, p)$. $\forall a > p$, find large deviation bound for $P( S \geq an)$ In the book, the large deviation bound definition is as follows: $\phi(t)$ is finite for some $t > 0$, ...
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2answers
31 views

Markov chain - Stationnary distribution - Unique

Consider the following respective infinitesimal generators of Markov chains in continuous time: \begin{equation} A=\begin{bmatrix} -4 & 1 & 3 \\ 3 & -5 & 2 \\\ 0 & 3 ...
1
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1answer
18 views

What is your expected waiting time if limousine inter-departure times follow an exponential distribution?

Limousines depart from the railway station to the airport from the early morning till late at night. The limousines leave from the railway station with independent inter-departure times that are ...
0
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0answers
34 views

Probability of finding $n$ individuals in the logistic model

A population has a birth rate proportional to both the actual population, and its difference with a certain saturation population $\sigma$. The equation for the probability of finding $n$ individuals ...
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0answers
20 views

Generating a predictive distribution from a small sample

I have an oracle that gives numbers from an approximately-normal distribution, but I do not know the mean or variance of the distribution itself. (I do happen to that the oracle does not produce ...
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1answer
15 views

Initial point and initial distribution of the Markov chains

I am reading about Markov chains on a general state space and the ergodicity theory. Some of the ergodic theorems are presented when we consider n-step transition probability conditional on initial ...
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0answers
20 views

The MGF of a PDF confusion

A random variable has a pdf given by $$ f(x) = \left\{ \begin{array}{ll} x & \quad 0 \leq x<1 \\ 2-x & \quad 1 \leq x < 2 \end{array} \right. ...
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0answers
11 views

Entropy of degenerate multivariate normal

After having read the Wikipedia entry on the degenerate case of the multivariate normal distribution: https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Degenerate_case my question is: ...
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0answers
13 views

Is there such a thing as a Negative Poisson Binomial?

The Poisson Binomial distribution is a generalization of the binomial for non-equiprobable Bernoulli trials. The negative Binomial give the number of Bernoulli trials until a number of success is ...
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1answer
69 views

Can the partial sums of independent random variables with no normalization converge in distribution to a constant?

If $\{X_n , n\ge 1 \}$ is a sequence of independent random variables and $X_n$ is nondegenerate for at least one $n\ge1$, can there exist a finite constant c such that $S_n = \sum_{j=1}^n X_j ...
0
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1answer
21 views

Let $X ∼ E(2)$. Find the density function $f_Y$ of $Y = X^3$ .

Let $X ∼ E(2)$. Find the density function $f_Y$ of $Y = X^3$. Anyway I could get help starting out on this problem? I'm stumped as to how to approach this.
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0answers
13 views

polynomial chaos expansion: linear combination properties?

I'm dealing with polynomial chaos expansion, for finite support specifically. Assume $X$ and $Y$ are r.v.'s whose the inverse CDFs expressed as $$ F^{-1}_X(x) = \sum_{j=0}^{N} s_j^{(X)} \psi(\xi)$$ ...
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0answers
15 views

Using CLT to determine the sample size to achieve a given power.

Consider a distribution having a pmf of the form $f(x;\theta)=\theta^x(1-\theta)^{1-x}$ $x=0,1$, zero elsewehre. Let $H_0: \theta=\frac{1}{20}$ and $H_1: \theta>\frac{1}{20}$. Use the Central Limit ...
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0answers
16 views

The quotient of two chi distributions

The quotient distribution of two chi-squared distributions is F-distribution. What would be the quotient distribution of two chi distributions? Is there a general distribution for this?
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0answers
25 views

Models for Probability Density Functions with unknown parameters and given mean and variance

The PDF $f(x)$ of a non-negative random variable $x$ has the structure $$f(x)=\exp (a-bx-cx^{2})$$ where $a$, $b$ and $c$ are any model parameters. It is assumed that $c\ge 0$ so that $f(x)$ does not ...
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1answer
18 views

Bernoulli distribution with non integer number of trials

Can we generalise the Binomial distribution for a non-integer number of Bernoulli trials?
1
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1answer
38 views

finding the marginal density of Y

Question . The joint probability density function of X and Y is given by $f(x, y) = (1/8)(y^2 − x^2)e^{-y} , -y\leq x\leq y, 0\leq y \leq \infty $ Find the marginal density of x. So i know that we ...
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0answers
61 views

Dependence (Joint probability distribution, marginal distribution, unconditional and conditional mean, variance) [closed]

Let Y be a random variable defined as the sum of 5 independent Bernoulli trials in which the probability of each Bernoulli taking the value 1 is given by r. Suppose that prior to the 5 Bernoulli ...
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1answer
73 views

Probability that at least two of three uniform random variables~[0 1.5] add up to >2 [closed]

There's a problem I've been stuck on for a while regarding the sum of two uniformly distributed, independent random variables. The problem goes like this: You find some old batteries in a drawer. ...
1
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1answer
19 views

Distribution of linear combination of discrete variables

Assume $X,Y$ are discrete independent random variables with known distribution $P_X(x), P_Y(y)$ and $c_1, c_2$ constants. Can we determine the shape of the distribution of: $Z = c_1~X+c_2~Y$
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1answer
17 views

Sum of two random variables converging with different modes [closed]

Is it true that if X_n converges in distribution to X; Y_n converges in probability to Y; X_n, Y_n, X and Y are real-valued random variables defined on the same probability space, then X_n + Y_n ...
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0answers
11 views

Is there a tool/program out there that allows you to draw probability function and it spits the equation, runs simulations, etc…?

Is there a tool that allows you to draw a probability function (with some parameters pre-set such as the # of polynomials or curves, discrete/continuous, etc...) and the program: Spits out the ...
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0answers
21 views

Upper bound for most likely category in multinomial random vector NOT being max count realized

Let $(X_1,\dotsc, X_k)$ be distributed multinomial with parameters $n, (p_1\dotsc,p_k)$ and suppose $p_1>p_j$ for $j\neq 1$ so that category 1 is the most likely outcome from any given realization. ...
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0answers
25 views

how can I find the PDF of g(x) when the Characteristic function is known?

Suppose that the characteristic function of X is given ($sin(\alpha\omega/2 \pi)$ ) how can I find the PDF of the $y=x^2$ ?I think we should find the PDF of the function X (using the related ...
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1answer
28 views

Unknown distribution of a random variable

$X_1, X_2, \ldots, X_{400}$ is a random sample from given distribution with median of m ($P(X_i \le m)=0.5$). Calculate $P(X_{220:400} \le m)$. How to calculate that? I am lost with this question. ...
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1answer
22 views

What is the average number of random selections it would take to have picked every element of a set and the size of that set, n?

I've been discussing this question with my AP statistics teacher and we're both racking our brains as to how this probability distribution would look. The problem came up when looking at the scenario ...
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0answers
21 views

Find EX VX and sum of these function are 1? [closed]

![problem][1] It is simillar to binomial distribution`s probability mass function but I g
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0answers
16 views

Convergence in distribution of the following sequence of random variables

$X_n\sim Beta\left(\frac{\alpha}{n},\frac{\beta}{n}\right)$ with $\alpha>0$ and $\beta>0$. Does $X_n$ converge to a distribution?
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0answers
22 views

Max of independent Poisson random variables with widely different means

Suppose $0<\lambda_1\leq \lambda_2\leq \ldots\leq \lambda_k.$ For each integer $n>0,$ and $1\leq i\leq k,$ let random variable $X_{n,i}$ be distributed as a Poisson random variable with mean ...
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0answers
27 views

Probability density function with two peaks and skewness

I have plotted a probability density function on a graph. With one line from $(0,1)$ to $(1,0)$ and the second line from $(1,0)$ to $(2,1)$. The area under the lines sum up to $1$ and all values of ...
0
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1answer
27 views

memoryless property of exponential distributions with random variables

It is true that $P(X>t+s|X>t)=P(X>s)$ for certain values $t$ and $s$. However, how can I show that this still holds if: $T$ is a continuous random variable. That is ...
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0answers
25 views

Gap probability for i.i.d. random variables

Given a set $\{X_1,\ldots,X_N\}$ of real i.i.d. random variables, drawn from a common parent pdf $p_X(x)$, what is the probability that, given one random variable taking value in $(t-dt,t)$, there are ...
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1answer
22 views

Product of discrete random variable with constant

Say that we have a random variable $X$ and the distribution $P_X$. I know that is $X$ in continuous then, from the principle of conservation of probability, we get $P_{cX} = \frac 1 {|c|} P_X(\frac ...
0
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1answer
21 views

Why is chi-square distribution with 2 degrees of freedom an exponential distribution?

Is there any explanation on why these two distributions are equivalent? How can the sum of two square of Gaussians represents the limit of a geometric distribution? I found an answer here, which ...