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

learn more… | top users | synonyms

0
votes
0answers
16 views

Steps: How to derive Probability density function for geometric functions

I am not from Mathematics background and hence lack awareness of many basic knowledge. So, please pardon if this sounds too trivial. I would like to know the steps with which I can obtain the pdf of ...
0
votes
1answer
7 views

Beta-binomial random number generator

Could someone help me find a random number generator from a Beta-Binomial distribution in MATLAB, R or SAS? Thank you!
0
votes
1answer
22 views

density joint function

I got a question and I was stuck for more than 15 minutes... Here is the question, And the question was: Find F(1/2,2). I tried to reason but the answer was different from what I got, here is the ...
2
votes
1answer
12 views

Series of independent gaussian variables and brownian motion

I am checking the proof of the construction of a brownian motion in $[0,\pi]$. We show that \begin{gather*} t \mapsto B^m_t = \frac{t}{\sqrt{\pi}}X_0 + \sqrt{\frac{2}{\pi}}\sum_{n=1}^{2^m-1}X_n ...
0
votes
0answers
14 views

iid negative binomial conditional on the sum. Expectation and Variance

Hi and thanks for reading this in advance. I am mostly interested in expected value and variance but also the distribution of iid negative Binomial RVs conditional on there sum. I worked out the form ...
3
votes
0answers
22 views

Generalized Binomial Model independent in the limit

Start with a generalized binomial model $$P(X_{n+1}=1\mid \mathcal{F}_n)=\theta_n+ n^{-1} d_n \sum_{i=1}^n X_i$$ $$P(X_{n+1}=1)=p_{n+1}=\theta_n + n^{-1}d_n \sum_{i=1}^n p_i$$ Then if we impose ...
3
votes
0answers
21 views

Probability density function of $x$ in the unit circle?

I'm trying to work out how to find the probability density function (PDF) for $x$ values on the unit circle - not within the unit circle but on the edge. The reason for doing so is that I'm trying to ...
-2
votes
1answer
30 views

How central limit theorem invoke two random variables jointly Gaussian or independent? [on hold]

$X$=$\sum_{i=1}^n d_icos\theta_i$ and $Y$=$\sum_{i=1}^n d_isin\theta_i$ If n is large enough, and $d_i$ and $\theta_i$ are independent, then X and Y are jointly Gaussian? Is it ture? and if is please ...
0
votes
2answers
34 views

Express the CDF of $Y=X^2$

Let $X$ be a random variable with CDF $F$. Express the CDF of $Y=X^2$ in terms of $F$.
0
votes
1answer
27 views

Conditional Distribution: how to set up Limit of Integration of a joint density

I have a question in conditional probability. I'm asked to find the conditional distribution, however, I'm unsure about the answer given and would appreciate someone helping straighten out the theory ...
0
votes
0answers
21 views

How to develop a probability distribution/density function of an issue?

Assume that a health insurance company has $1000$ customers. It is estimated that the probability of a customer making a claim is $p = 0.2$ per year, independently of previous claims and other ...
0
votes
1answer
32 views

Random variable related to binomial

The number of successes $A$ in $n$ independent trials with the probability of a success is $p$ for each trial is binomially-distributed. I am interested in a scenario that adds dependence to the ...
-1
votes
2answers
71 views

What is the distribution of Z=min(X,Y) [on hold]

Let X and Y be independent geometric random variables. What is the distribution of Z=min(X,Y)?
0
votes
1answer
12 views

Show that the σ-algebras generated by the collection of all intervals are equivalent

Show that the σ-algebras generated by the collection of all intervals of the form [a,b]⊂R and by the collection of all the intervals of the form (−∞,b]⊂R are equivalent. i am having trouble with ...
0
votes
0answers
5 views

Use copula to find the joint distribution of two random variables

Assuming that there are two random variables $x$, $y$ both having an Arcsine marginal distribution $F$, i.e. $F(x)=\frac{2}{\pi}\arcsin{\sqrt{x}}$ The density of their Gaussian copula can thus be ...
0
votes
0answers
12 views

Existence of independent standard normally distributed r.v.s

For any standard normally distributed random variable $X$, show that we can find another standard normally distributed random variable $Y$ that is independent of $X$. This is one of problems in my ...
1
vote
1answer
14 views

Combined Distribution of Random variable

How to compute $P[T1 \le T2 \le t]$ for T1, T2 is independent random variable with exponential distribution in terms of cmf, pdf of T1 and T2? Similarly for $P[T1 \le T2 \le T3.. \le t]$ ? I tried ...
0
votes
2answers
23 views

Find the probability density function of $Y=X^2$

Consider the random variable X with probability density function $$f(x)=3x^2$$ if $0<x<1$, and $$f(x)=0$$ otherwise. Find the probability density function of $Y=X^2$. This is the first question ...
1
vote
0answers
15 views

CDF of maximum of iid rvs [duplicate]

I am having a small doubt regarding maximum of random variables. I have $$Z= \max\{ X_1, X_2,\dots X_p, \dots X_N\}$$ where all $X_i$ are independent, identically distributed. Now, If for sure, I know ...
0
votes
0answers
16 views

Bernoulli random variable uniformly distributed have same distribution as a subsequent binomial variable? [on hold]

Suppose two Bernoulli random variables X1 and X2 are individually uniformly distributed. Is Y= X1 + X2 therefore also uniformly distributed?
1
vote
1answer
21 views

create a Gaussian distribution with a customize covariance in Matlab

the Matlab function 'randn' randomize a Gaussian distribution with $\mu= \begin {pmatrix} 0\\0\end{pmatrix}$ and $cov= \begin {pmatrix} 1&0\\0&1\end{pmatrix}$ Ineed to randomize a Gaussian ...
1
vote
1answer
23 views

Given a probability distribution, how many times do I have to repeat an experiment so see a certain outcome

My question concerns random number generation under certain constraints. I assume that the random number generator is good enough to generate uniformly distributed numbers. This means that each number ...
1
vote
1answer
37 views

A generalization of the Glivenko-Cantelli theorem

Let $P$ and $P_n$ be probability measures on $\mathcal{B}(\mathbb{R})$ with distribution functions $F$ and $F_n$. Moreover, let $F$ be continuous and $(P_n)_{n\in\mathbb{N}}$ weakly converge to $P$. ...
0
votes
0answers
18 views

How to solve $P(Y_1 ≤ 3/4, Y_2 ≥ 1/2)$. Pls help

Let $Y_1$ and $Y_2$ have the joint probability density function given by $$f (y_1, y_2) =k(1 − y_2), 0 ≤ y_1 ≤ y_2 ≤ 1$$ $$ 0,\,\,\,\, elsewhere.$$ a) Find the value of $k$ that makes this a ...
0
votes
0answers
6 views

Signal-extraction knowing both the sum and the sum of the absolute values of normally distributed variables

I have two normally distributed variables $X∼N(μ_{x},σ_{x}²)$ and $Y∼N(μ_{y},σ_{y}²)$. I can observe both the sum of their values and the sum of their absolute values, i.e. $Z₁=X+Y$ and $Z₂=|X|+|Y|$. ...
0
votes
0answers
20 views

Dynamic programming for optimal maximum and optimal minimum

We have a sequence of $a_i$ and a choosing rule that is take the first number $x_t\ge a_t$. The definition is = $$ min\{ t|t \in \{ 1,2,\cdots,n\}\,\,,\,\, x_t\ge a_t\}$$ The sequence $a_i$ is ...
1
vote
1answer
26 views

Sum of Two Poisson distributions

The probability distribution for the number of goals scored per match by Team A is believed to follow $X \sim Poi(0.8)$. Independently, the number of goals scored by Team B is believed to ...
2
votes
1answer
38 views

Joint probability distribution

$Y_1$ and $Y_2$ are jointly distributed with density $f(y_1,y_2)=4y_2^2 \qquad 0 \leq y_1 \leq y_2 \leq 1$ Determine the following: $P(\text{max} \{Y_1,Y_2\} <1/2) = ...
1
vote
0answers
38 views

Let $X_n>0$ be iid and $P(X_n>t)\sim t^{-\alpha}$, show that $Y_n=n^{-1/\alpha}S_n$ and $1/Y_n$ are tight.

We are given that $X_n>0$ be iid with common distribtuon $X$, and $P(X>t)\sim t^{-\alpha}$, I need to show that the scale of $Y_n$ is $n^{1/\alpha}$. Or in other words show that ...
1
vote
2answers
21 views

Support of the conditional distribution of a poisson process

I am working on Problem 5.1.8 of this book. It states: Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the conditional distribution of ...
1
vote
0answers
57 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the ...
-1
votes
2answers
22 views

Probability of the highest order statistic below the population median.

What is the probability that the highest order statistic of a random sample of size n from any continuous distribution is below the median ( population median ) of that distribution.
-2
votes
1answer
22 views

Probability: How much days we need to play a game win

Suppose the probability of win a lotery game is : $1/1000$ If a person play the lotery every day with the same combination, how much time he need to wait to win the lotery? Im thinking to use a ...
0
votes
1answer
18 views

Multivariate sampling of $F(x_1,…,x_n)$?

Let $$(X_1,...,X_n)\sim F(x_1,...,x_n)$$ (not independent). How can I sample from this distribution? In the univariate case, on can use $F^{-1}(u),u\sim U(0,1)$. However, in the multivariate case ...
1
vote
2answers
67 views

Prove that if $X$ is stochastically larger than $Y$ then $E(X)\ge E(Y)$

Prove that if $X$ is stochastically larger than $Y$ (i.e. $P(X > t) \ge P(Y > t)$ then $E(X)\ge E(Y)$. I understand how to solve the problem if $X$ and $Y$ are non-negative random ...
1
vote
0answers
32 views

A property of the hazard function of the normal distribution

I have a problem that I can't figure out. Define $$\Gamma\left(x\right):=\frac{\phi(x)}{1-\Phi(x)}$$ where $\phi(x)$, $\Phi(x)$ are the density respectively cumulative distribution function of the ...
1
vote
0answers
12 views

question about exponential distribution or exponential random variables

Consider a post office that is run by two clerks. Suppose that when Mr. Anderson enters the system he discovers that Mr. Smith is being served by clerk 1 and Mr. Brown by clerk 2. Suppose also that ...
1
vote
2answers
62 views

Show that Y=aX+b is an random variable.

Let X be an random variable on a given probability space and let a,b∈R. Show that Y=aX+b is an random variable. if X has a distribution function F, what is the distribution function of Y? if X ...
0
votes
0answers
15 views

What is the limiting distribution of this Markov Chain?

Take a Markov Chain with state space $\left\{ 0, 1, \dots, 20 \right\}$. Then we have the rule that given $X_n$: Compute $Z = X_n + 1$ or $Z = X_n - 1$ with probability $\frac{1}{2}$ each (if the ...
1
vote
0answers
10 views

4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf z} \in C^{n×1}$ is a CSCG random vector denoted with $C (μ,Σ)$ where $μ$ and $Σ$ are mean and contrivance matrix, respectively, and defined as $μ=E({\bf z})$, $Σ=E({\bf z}{\bf ...
0
votes
3answers
31 views

A box contains 5 yellow and 3 red balls, from which 4 balls are drawn one at a time, at random, without replacement.

A box contains 5 yellow and 3 red balls, from which 4 balls are drawn one at a time, at random, without replacement. Let $X$ be the number of yellow balls on the first two draws and $Y$ the number of ...
1
vote
0answers
13 views

Simulate from a distribution using Metropolis-Hastings and Rejection Sampling?

We have covered the basics behind rejection sampling as well as Metropolis-Hastings from class, but I am not sure how to use the two in conjunction to solve the following problem: Given $\pi(x) = ...
1
vote
1answer
34 views

Mean of Poisson distribution

Let $X$ have a Poisson distribution with double mode at $x=1$ and $x=2$. Find $ P(x=0)$.Here is my solution... $\mu= \frac {p(2) 2!}{p(1)}$. then how can find the mean..thanks
0
votes
1answer
24 views

Find the cummulative distribution function and the density function of the random variable: $Y={1\over 1+U}$

Let $U$ have a uniform distribution on $[0,1]$. Find the cummulative distribution function and the density function of the random variable: $Y={1\over 1+U}$ My attempt: $F_Y(x)=P[Y\le x]=P[{1\over ...
1
vote
0answers
7 views

What is the transformation that maps a Gaussian distribution to a Beta distribution?

Suppose X is a random variable with Gaussian distribution over domain $\mathbb{R} = (-\infty, +\infty)$, with PDF function $f_X$. And Y is a random variable with Beta distribution over domain ...
1
vote
1answer
30 views

Probability of Sample Variance Given Variance

I am trying to solve a problem that I have never seen before and cant seem to find a way to solve it so any help or tips would be appreciated! Here's the Problem: Suppose a considerable amount of ...
1
vote
0answers
23 views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P(F^{-1}(F(X)) ...
0
votes
1answer
36 views

Shortcut to finding $E(XY)$

The question says "Find $E(Y|X)$ and hence evaluate $E(Y)$ and $E(XY)$" The joint pdf is $$f_{X,Y}(x,y)=\begin{cases} 8xy, & \text{ for } 0< y< x < 1, \\0, & \text{ elsewhere } ...
1
vote
1answer
13 views

Distributing dimes to 2 groups of people such that each member of one group gets at least one

I have a study question that I have the answer for, but I just can't understand how or why it is the answer. The question is: $n$ dimes are distributed to $y$ young people and $o$ old people. Every ...
2
votes
0answers
15 views

Prove $|\log F(v)|\leq |\log F(0)|+|v|+|v|^2$ for $F$ is the standard normal CDF

Suppose that $F$ is the CDF of a standard normal distribution. Hayashi (2000) claims that the following is true $$ |\log F(v)|\leq |\log F(0)|+|v|+|v|^2\quad\text{for all}\quad v. $$ How does one ...