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

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1answer
46 views

Is there a shorter way for me to answer this joint probability question?

Suppose a box contains 10 green, 10 red, and 10 black balls. We draw 10 balls from the box by sampling with replacement. Let X be the number of green balls, and Y be the number of black balls in the ...
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0answers
28 views

Simulating beta random variables

Let's say I estimated the parameters of Beta distribution with a single-period dataset, and want to generate a multi-period sample. How can I do that? To be more specific, I estimate my parameters ...
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1answer
36 views

Finding the CDF of $g(X)$ where $X$ is a continuous random variable

I imagine this is a rather simple problem, but I'm having a bit of a hard time actually finding the answer. $X \sim \mathrm{Exp}(0.2)$ and $W=g(X)$ given by $g(X) = \begin{cases} X^{\frac{1}{3}} ...
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1answer
39 views

Derivative of integral over part of Gaussian distribution

I am currently trying to compute the following derivative and integral: $$ P\psi_\theta = \frac{d}{d\theta}\int_{-k}^k tf_T(t)dt, $$ where $t=x-\theta$ and $X\sim N(\theta_0,\sigma^2)$. $f_T$ above ...
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2answers
224 views

Defining a probability density function in R (software), and sampling from it

I'm asked to generate a random sample from a logistic distribution with the PDF, $$f(x)=\frac{e^{-x}}{(1+e^{-x})^2}$$ without using the function rlogis(...) So I ...
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1answer
25 views

Two random variables X,Y are, X,Y independant b. are X+Y X-Y independant

if X and Y are independent, check whether the (0, 0) value is the same as P(X=0) P(Y= 0), and the same with the other 4 entries. Make a table with the distributions of X + Y and X - Y. For any ...
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2answers
75 views

Continuous and Discrete random variable distribution function

I have a very basic question in probability. It pertains to the difference between a continuous random variable distribution function and a discrete one. This question has confused me many times. ...
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1answer
27 views

Variance from the pdf

FOr the interpretation of the variance, it is the fluctuation of the data around the mean. So if I know that mean (say mean=0), and then there are lots of data (70%) points are greater than +/-10 away ...
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1answer
158 views

Explanation of how probability density functions transform under the change of variable

I've just read about probability density function from this article. In that article, there is some wired concept that I can't understand, please see the section named "Dependent variables and change ...
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4answers
36 views

Proving consequence of $\operatorname{var}(X)+\operatorname{var}(Y)=\operatorname{var}(X+Y)$

How to prove that if $\operatorname{var}(X)+\operatorname{var}(Y)=\operatorname{var}(X+Y)$, then $X$ and $Y$ are independent?
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1answer
63 views

Probability: basic question and concept

I have always been struggling with the problem, in particular, I usually have great difficulty in differenting when should I multiply n! to take care of the ordering, and when should I not do so. For ...
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0answers
52 views

Differential Equation for brownian bridge?

For the brownian motion, we know that probability density of the particle's position at time $ t $, $ \rho(x,t) $ satisfies the diffusion equation pde: $ \partial_t \rho = d \; \partial_x^2 \rho $. Is ...
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6answers
249 views

How to show this integral equals $\pi^2$?

I was trying to evaluate an integral related to the product of two cauchy distributions and in one of the steps got stuck in the integral $$\int_0^{\infty} \frac{\ln(x)}{\sqrt{x}(x-1)} dx. $$ I ...
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1answer
54 views

Lifetime of Light Bulbs - Probability Question

This is the question that I have, so I solved the first two parts very easily. The first part (i) Then the part (ii) Now, I dont know how to do the final part of the question (it is too ...
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1answer
98 views

How to Bayes update discontinuous cdf

I would like to find the probability of an event given a signal (update using Bayes' rule). The events are $x\in \mathbb{R}$ with pdf $g:\mathbb{R}\rightarrow \mathbb{R}_+$, with $g(x)>0\;\forall ...
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0answers
14 views

Methods for Uncorrelating data - Comparison

I see that both PCA and Cholesky Decomposition could be used for uncorrelating correlated data. When should one be used? What are the assumptions made by each model. When do the methods fail? Are ...
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0answers
51 views

Distributed to symbol for frequency distributions

How would you write down that some random variable $X$ is distributed to a frequency distribution. For the normal distribution e.g. I often see sth. like that: $X\sim \mathcal{N}(\mu,\sigma^2)$. Is ...
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2answers
45 views

What's the PMF of the Difference between 2 Independent Poisson RV?

I searched around and found that the difference between 2 independent Poisson RV $X_1$ (mean $\mu_1$) and $X_2$ (mean $\mu_2$) follows the Skellam distribution such that its PMF is: $$f(k; \mu_1, ...
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1answer
79 views

Distribution of the sample variance of n iid exponential variables

I have to check some properties of an estimator, but I can't find its distribution. Let $X_1,...,X_n $ be independent identically distributed exponential variables with parameter $ \theta $, i.e. ...
2
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1answer
297 views

Throwing darts at dartboard (cumulative distribution function)

Suppose there is a target shooting game on circle of radius $3$. Think of the result of the shooting as a random experiment, for simplicity, we suppose the hit will always impact on the circle of ...
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0answers
31 views

Finding $a_n, b_n$ so that a sequence converges in distribution to a nondegenerate random variable.

Now, $X_1, X_2,\dots$ are iid with the same distribution as the chi-squared distribution with one degree of freedom. Find $a_n$ and $b_n$ so that $a_n \left( \max_{1 \leq i \leq n} X_i - b_n \right)$ ...
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1answer
13 views

$U\sim \mathcal U(0,a)\overset{?}{\implies}U-\lfloor U \rfloor \sim \mathcal U (0,a-\lfloor a \rfloor$)

Suppose $U\sim \mathcal U(0,a)$ for some $a>0$. Is it true that $U-\lfloor U \rfloor \sim \mathcal U (0,a-\lfloor a \rfloor$)? How can I prove this? If $a\in \mathbb N$ then the following ...
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2answers
323 views

uniform distribution over disk

Given two independent random variables $A$ uniform on $[0,1]$ and $B$ uniform on $[0,2\pi]$. Obtain the joint pdf, tranform to the disk, if necessary modify to obtain the uniform pdf over the disk. ...
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2answers
43 views

What does tightness of convergence mean and why do we use this?

In a paper that I am reading it is written: We have $$(X(t_j))_{j=1}^k \xrightarrow{d} (Y(t_j))_{j=1}^k,$$ where $t_0=0, t_1 < \dotsb < t_k$. It further is written: In order to show that ...
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0answers
90 views

Upper bound on the expectation $E[(X-t)^+]$

Let $X$ be a random variable with $E[X]=0$ and $E[X^2]=1$ that satisfies \begin{equation} |F_X(x) - \Phi(x)| \leq \alpha, \qquad \forall x\in\mathbb{R}, \end{equation} where $F_X(\cdot)$ is ...
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0answers
32 views

Conditions for the solution to a matrix differential equation (with exponentials) to be a valid probability distribution

Define $F(z) : \mathbb{R} \to [0,1]^2$, $C \in \mathbb{R}^{2 \times 2}$ and $D \in \mathbb{R}^{2}$. The scalar $z \in [0, \infty)$ A few further assumptions: $C$ is negative definite and ...
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0answers
37 views

Marginal of Multivariate Mixture of Gaussians

I have a multivariate mixture of gaussians with $K$ components of $N$ dimensions. I would like to collapse the mixture into one of lower dimension by computing the marginal for the first part, $x_A$, ...
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0answers
44 views

Expectation of $\frac{1}{X+1}$ for a geometric random variable

I am confused over $E(\frac{1}{1+X})$ where $X$ is geometric distribution with parameter $p$. The book wants me to prove that $E(\frac{1}{1+X})=log((1-p)^{\frac{p}{p-1}})$ Here's what I did. ...
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2answers
244 views

Determine the PDF from the MGF

If the moment generating function is given as; $ \psi_X(s) = e^{s^2}$ How can i determine the PDF of $X$?
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17 views

Presenting a multinomial dstribution as some function of underlying binomials

I have a multinomial distribution, which arises, let's say, for the sake of clarity, from $N$ rolls of unfair $S$ sided dice and labels on the sides are non-integer. I know the probability for each ...
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1answer
22 views

Testing statistic $\frac{MSS(X)}{MSS(Y)}$

Suppose a test statistic $\frac{MSS(X)}{MSS(Y)}$, where $MSS$ denotes Mean Sum of Squares, is to be used for testing the significance of the factor $X$. Do we need the assumption $$\mathbb ...
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2answers
46 views

Correlation of random variables with joint PDF proportional to $x^{a-1}y^{b-1}(1-x-y)^{c-1} $

The random variables $X$ and $Y$ have joint PDF $$f(x,y)= \frac{\Gamma(a+b+c)}{\Gamma(a)\Gamma(b)\Gamma(c)}x^{a-1}y^{b-1}(1-x-y)^{c-1} $$ where $0 \leq x \leq 1 , 0 \leq y \leq 1, x+y < 1 $ where ...
1
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1answer
88 views

What is the correct equation for conditional relative entropy and why

I was trying to understand the concept of conditional relative entropy. As in: $$D(P(X\mid Y) ||Q(X\mid Y))= E [\log\frac{P(X\mid Y)}{Q(X\mid Y)}]$$ I would have thought that its equations would ...
3
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1answer
109 views

Optimization with probability densities - Lagrange multipliers

This question is concerned with the paper "A Lower Bound for a Probability Moment of any Absolutely Continuous Distribution with Finite Variance" by Sigeiti Moriguti appeared in Ann. Math. Statist. ...
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0answers
41 views

A decisive test to tell if a function is separable

This is a both physics and math question. I have a Hamiltonian in the form H= q1^2 + q2^2 + q3^2 + q1 x q2 x q3 Therefore the probability distribution will ...
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0answers
250 views

Distribution of the sum of many lognormal random numbers from same distribution

In my application I have to sum up a lot (between 1000 and 2000) lognormally distributed random numbers and use their sum. All random numbers that I sum up follow the same distribution. The current ...
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1answer
113 views

Expected Value of a Minimum Function using a Beta Distribution

Let $X$ be a IID random variable with support in $[0,1]$ and CDF given by a Beta distribution, i.e. $X \sim Beta(\alpha,1)$. Suppose we have a function of the form: $$ Z_t = \phi(X_t,y_{t-1}) = ...
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3answers
53 views

Compute variance, using explicit PDF

I'm trying to get $\text{Var}(x)$ of $f(x) = 2(1+x)^{-3},\ x>0$. Please tell me if my working is correct and/or whether there is a better method I can use to get this more easily. $$ ...
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1answer
99 views

finding out the probability density of a random process

I have to find out the probability density function of a random process with the following specifications:z(t)= xcos(wt)-ysin(wt) where x and y are two independent gaussian random variables. Now what ...
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1answer
31 views

Poisson distribution equation

This is probably a very simple and silly question to ask, but I just don't understand the steps for b). I don't quite understand where the negative (-) sign came from? Could somebody please shed some ...
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1answer
22 views

What is a time of waiting for 5th success in bernoulli's sequence with p - probability.

What is a time of waiting for 5th success in bernoulli's sequence with p - probability. Hum, what exactly should I found? Should I use Newton distribution for r=5, but what is my k?
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0answers
83 views

Cumulative distribution function for a Poisson distribution

This is a past exam question and I just want your opinions on if it's sufficient or not. I had to prove: Let the discrete random variable $X$ have a Poisson distribution with parameter $\lambda$. ...
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2answers
49 views

Help finding k. Issue with integration

Let the continuous random variable $X$ have a probability density function $f(x)$ such that $$f(x) = k(1+x)^{-3}, x>0$$ $=0$ elsewhere Find k This is what I tried: $\int_0^\infty k(1+x)^{-3}dx ...
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0answers
63 views

Integration of a continuous function under Lebesgue-Stieltjes measure space using simple functions

I am struggling to prove the following result using an approximating sequence of simple functions. Could anyone give me a clue? Under a Lebesgue-Stieltjes measure space ...
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2answers
99 views

Given a CDF, find P(-.5<X<.5)

Given the following CDF: \begin{equation*} F(x)= \left\{ \begin{array}{lr} 0 & x<-1, \\ \frac{x+2}{4} & -1 \leq x < 1 \\ 1 & x \leq 1 \end{array} \right. \end{equation*} Compute ...
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3answers
168 views

Some issues concerning joint random variables

Let the joint random variable $P[x;y]$ be $P[x;y] = c[2x^2 + y^2], x=-1;0;1, y=1;2;3;4$ $=0$ $elsewhere$ So I had to find the value of $c$ that makes $P[x;y]$ a joint discrete random variable. I ...
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25 views

applicability of Poisson distribution

Do all random variables dealing with number of random, independent events in a continuous interval follow a Poisson distribution?
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60 views

Finding the value of c for which two probabilities are equal

May I please get help with this question? The amount of a certain chemical in a type A cell is normally distributed with mean of 10 and a standard deviation of 1, while the amount in a type B cell is ...
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0answers
40 views

Computing the expectation of following interesting random variable

Every package of some cereal includes a plastic animal. There are $N$ different types of animals, and each package is equally likely to contain any type. Your children make you buy one package of the ...
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2answers
154 views

Expectation of maximum of iid random variables

Let $X_1, X_2, \ldots, X_n$ be independent random variables having the common density function $f(x)$. We have $$f(x) = \begin{cases} 1 & \text{for } 0 < x < 1, \\ 0 & \text{otherwise} ...