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

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125 views

Quick Question Integration with Joint PDF

Let $X_1, X_2, \ldots, X_n$ by independent and identically distributed random variables with probability density function (pdf) $$f_X(x) = \left\{\begin{array}{ll}1, & 0 < x < 1\\ 0, ...
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1answer
35 views

Find marginal and conditional distributions [closed]

Consider the probabiility density function $f_{X_1, X_2}(x_1, x_2) = \left\{\begin{matrix}\frac{1}{8x_2} \exp\left\{ -\left( \frac{x_1}{2x_2} + \frac{x_2}{4}\right)\right\}, & x_1 > 0, ...
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1answer
19 views

Expectation of minimum set of i.i.d random stopping times with the same distribution

What is the expectation of the minimum set of n i.i.d random stopping times? is it \frac{T}{n}
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1answer
105 views

Marginal PDF with dependent variables

I don't understand how to work out the limits of integration in b).
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1answer
36 views

Is this a Markov chain property

For $A,B$ measurable sets and $(X_n)_n$ a Markov chain. Do any of the following properties hold? $$P(X_2 \in B | X_1=x_1,X_0 \in A) = P(X_2 \in B|X_1=x_1)$$ or $$P(X_2 \in B|X_1 \in A,X_0=x_0) = ...
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0answers
115 views

Moment-generating function of a generalised normal random variable

Let $X$ be a random variable that follows the "version 1" generalised normal distribution described here, with p.d.f. ...
2
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1answer
89 views

Urn with balls, distribution of random variable

From an urn containing $6$ balls numerated $1,\ldots,6$ we randomly choose one, then again and stop only when we picked the ball with number $1$ on it. Let $X$ be the greatest number that appeared on ...
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2answers
37 views

Gamma and Exponential distribution question?

The working time of one bank has an exponential distribution with a parameter λ=0.1 (in minutes). You came in the bank, but there were already 35 people before you. What's the probability that all of ...
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1answer
73 views

Distribution of minimum of two uniforms given the maximum

Let $X_1$ and $X_2$ be two random variables uniformly distributed on $(0, 1)$. It is easy to calculate the distribution of minimum and maximum of these two numbers: $$ P[\max(X_1, X_2)<x] = x^2 $$ ...
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1answer
132 views

Reability of any CDF in Excel based on the binomial one as Cumfreq does.

I'm trying to get my own excel sheet to calculate the confidence limits or belts. I'm interesting in apply it to the Two Components Extreme Values (TCEV) Distribution for Flood Frequency Analysis and ...
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2answers
95 views

What is the pdf of $X,Y$?

We know that the common pdf of $X,Y$ is constant function, on the triangle $(0,0),(0,1),(2,0)$ (and out of this range the value of the function is zero). What is $f_X(x)$ and $f_Y(y)$? My solution: ...
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1answer
88 views

Probability Estimator

Hi I was going through the MIT 2005 Machine Learning homework assignments and I was having trouble understanding a few concepts in probability theory. I would be obliged if anyone could validate my ...
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0answers
60 views

How to calculate total probability from independent events

Assume Y is caused by two independent events A & B, upon investigating a data set carrying 1000 entries we see. $$\begin{align}\text{Number of occurrence of events } A = 497 \text{ and } ...
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4answers
55 views

Verification of this summation [closed]

How do I check or evaluate this summation $$\sum_{k\ge 0} \left(\frac12\right)^{k+1}k=1$$
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1answer
45 views

Check or Evaluate this Summation

How do I check or evaluate this summation$$\sum_{k=0}^n \frac{2(k+1)}{(n+1)(n+2)}=1$$ for $0\le k\le n$
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1answer
120 views

Solution to a certain moment problem

I'm looking for a function $f$ that satisfies $f(x)\geq0$ $\int f(x) \mathrm{d}x=1$ $\int xf(x) \mathrm{d}x=0$ $\int x^2f(x)\mathrm{d}x=1$ $\int x^4f(x)\mathrm{d}x=\delta$ $\int ...
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2answers
63 views

Binomial/Negative Binomial Distribution? Why not Poisson here?

When I looked at the below problem, I thought of Poisson immediately. I converted the rate to making 9/10 shots. However the answer told me to use the binomial/negative binomial distribution for parts ...
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2answers
47 views

Joint distribution: show the components of the joint distribution are independent.

Very odd question I think... Show that if $(X,Y)$ is a random vector in $\mathbb{R}^{2}$ with density $f_{(X,Y)}(x,y) = f(x)g(y)$ for a pair of non-negative functions $f$ and $g$, then $X$ has ...
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1answer
97 views

Conjugated priors (Pareto and Beta): Does this distribution have a name?

$$F_X(x)=\begin{cases} \quad\dfrac{\alpha}{\alpha+\theta}\left(\dfrac x\omega \right)^\theta &\text{ if } x<\omega \\ \\ 1-\dfrac{\theta}{\alpha+\theta}\left(\dfrac\omega x\right)^{\alpha} ...
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1answer
32 views

Distribution functions of a probability measure on a probability space $(\mathbb{R},\mathcal{B})$

Let $F$ denote a distribution function of a probability measure $P$ on a probability space $(\mathbb{R},\mathcal{B})$, where $\mathcal{B}$ denotes the Borel $\sigma$-algebra on $\mathbb{R}$. Given ...
4
votes
1answer
434 views

Derivation of the negative hypergeometric distribution

Suppose we've given an urn which contains $R$ red and $W$ white balls. These balls are drawn randomly from the urn and are not placed back. Let $X:=$ number of attempts, before we've drawn at least ...
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1answer
23 views

Is there a function to tell if two probability vectors' max values are in the same dimension?

Is there a method or function to tell two probability vectors' max values are in the same dimension? Or Is there a bound for the angle of two normalized probability vector which their max values are ...
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3answers
47 views

Determine the Cumulative Distributive Distribution(CDF) of a truncated value?

It is the last part(part h) that I am having problems with. I know you use integration and then split it into 2 parts. But how exactly do you do it ? A detailed answer would be very helpful ! ...
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1answer
19 views

Bivariate Normal Probability

Assume we have a large data set of PSAT and SAT scores with bivariate normal distribution with $\rho = 0.6$. The mean and SD of the PSAT scores are (respectively) $1200$ and $100$. The mean and SD ...
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1answer
19 views

$\mathbb{P}(|X|<1,|Y|<2)$ When $X,Y$ Are I.I.D. Standard Normal

Calculate $\mathbb{P}(|X|<1,|Y|<2)$ when $X,Y$ are i.i.d. standard normal r.v.s. I think the answer is simply $$(\Phi(1)-\Phi(-1))(\Phi(2)-\Phi(-2)).$$ Is this correct? Thanks.
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2answers
1k views

Inverse gamma distribution

Wikipedia (at the time I write this) has two mutually inconsistent entries (one after the other !, http://en.wikipedia.org/wiki/Inverse-gamma_distribution#Properties): $$X \sim \mbox{Gamma}(k, ...
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1answer
60 views

Hypergeometric Distribution Confusion

I'm having trouble understand the part of the pmf for the Hypergeometric Distribution highlighted in green: $$\Pr[X = k] = \frac{\dbinom{m}{k}\!\!\color{green}{\dbinom{N-m}{n-k}}}{\dbinom{N}{n}}$$ ...
4
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1answer
131 views

Convegence of regularized sequence in $L^2$

Let $(\rho_n)_{n \geq 0}$ be a standard regularizing sequence on $\mathbb R$. Let $P$ be a probability measure on $\mathbb R$ such that the sequence $(P*\rho_n)_{n \geq 0}$ is bounded in $L^2$. Then, ...
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2answers
45 views

$X,Y\sim U(0,1)$ are independent, $W=\max (X,Y)$

$X,Y\sim U(0,1)$ are independent, $W=\max (X,Y)$. How do I find the PDF of $W$? How do I find the expectation of $W$ at two ways: 1. with the PDF of $W$ and without the PDF of $W$. I'd like to ...
3
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1answer
257 views

PDF of sum of two random variables

Assume an $n$ dimensional random variable $U$ that is uniformly distributed in the volume of an $n$-sphere with radius $R$. Assume another $n$ dimensional random variable $N$ that is distributed ...
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1answer
34 views

Gaussian distribution variance estimation

It's well known if I have a process generating normally distribuited data, I can estimate the parameters of the gaussian function: ...
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0answers
32 views

Rosanov - Probability Theory Chapter 4 Question 5

I am trying to solve one of the questions in Rosanov - Probability (Chapter 4 Question 5), but I am not exactly sure what the question is asking of me. The question is: Random variable $E$ with ...
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2answers
66 views

How many ways to represent a probability density function?

I have read accidentally in a book this sentence: " ... consider a random sample $X_1, X_2, \ldots, X_n$, each $X_i$ having probability distribution $f(x)dx$. Thus, we have $$\mathbb{P}(X_1\in ...
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1answer
229 views

Expectation and Variance of Poisson Process

Suppose that in a store, customers arrive as a Poisson process with rate $1/\mathrm{min}$ between time $0$ and $10$ minutes. Suppose there are ten kinds of items in the store (each kind has ...
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0answers
83 views

Expectation of Squared Non-Standard Normal

Let $X, Y, Z$ be independent standard normal variables. What is $\mathbb{E}(X+Y-Z)^2$? n.b.: this is review, not homework. I just wanted to double-check my own answer. Let $Q:=X+Y-Z$. ...
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1answer
1k views

Moment Generating Function of Gaussian Distribution

Derive from first principles, the moment generating function of a Gaussian Distribution with $$PDF= \dfrac{1}{\sqrt{2\pi \sigma^2}}e^{-(x- \mu)^2/2\sigma^2}$$ MY ATTEMPT MGF= E[$e^{tx}$]= ...
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0answers
33 views

Testing for a power law

How can we show wether or not a given probability distribution is a power law distribution? So for example it is know that a normal distribution is not a power law distribution where a student t ...
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0answers
48 views

Distribution of phone calls during 24h

I would like to model the amount of phone calls at each time of the day. The phone calls should follow a poisson distribution and at 12:00 there should be the peak. So, semantically what I would like ...
3
votes
1answer
49 views

What is the probability that $k$ events have occurred at time $t$, i.e., $\Pr[N(t)=k]$?

Assume that the starting time $T_0=0$. There are $n$ events that occur sequentially at time $T_1$, $T_2$, …, $T_n$, ($T_k\geqslant T_{k-1}$). Suppose the time intervals $\Delta{T_k}\,(\Delta{T_k} = ...
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0answers
44 views

For which joint distributions is a conditional expectation an additive function?

I know that, for a random vector $(X,Y,Z)$ jointly normally distributed, the conditional expectation $\mathbb{E}[\,X\mid Y=y,Z=z]$ is an additive function of $y$ and $z$. For what other distributions ...
0
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1answer
124 views

Evenly spreading study over 20 days?

Want to study $8$ hours a day, and $4$ hours on exam days. Want to study all exams exactly the same amount of time total. Exams in $10$,$13$,$19$,$20$ days from start. What is the daily ...
0
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2answers
112 views

Distribution of mean of Normal distribution

Suppose $X\sim N(\mu,\sigma)$. I want to find the following probability $P[\mu \ge \theta |x= \theta -c]$ for $c>0$. In another word, I saw a sample of Normal distribution, $x$, and know that it ...
0
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1answer
179 views

Conditional probability, Bernouilli's trial alternative method..

So I got this question on an practice test, and my first thought was to use Bernouilli's trial to solve for the answer. My question is how to possibly solve it without Bernouilli's trial, in a manner ...
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1answer
19 views

What is the prefered approach for this? (distribution)

Let's say I want to have a list of random numbers that follow a distribution. All random numbers should be between 0 and 100, and the mean is variable but doesn't change while we generate the randoms. ...
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1answer
27 views

Does correlation have to be in the context of (Gaussian) normal distribution?

I am not quite familiar with the concept of correlation. The Pearson's correlation coefficient is defined as: $\rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ...
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0answers
17 views

Rolling $n$ times with an $m$-sided dice. Closed, finite formula for the distribution of the sum? [duplicate]

My current idea is the following: practically we want to get the distribution of the sum of $n$-times of a discrete uniform distribution between $1,...,m$ . It is practically the discrete convolution ...
3
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1answer
214 views

Convergence of marginal distribtution

Here I have a question which looks a little bit weird: $(q_n)_n$ is sequence of probability density functions of the couple $(x,y) \in \mathbb R^2$, $p_n$ is the marginal density of $q_n$, i.e. ...
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2answers
136 views

Density function for a random variable having a mixed distribution

A random variable has the following mixed distribution (ie: A distribution that is both discrete and continuous): $P_{X}=\frac{1}{3}E(1)+\frac{2}{3}B(\frac{1}{2})$ Where E(1) is the exponential ...
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2answers
152 views

Probability that order statistic is larger than the other

Given the density function: $$f_Y(y)=e^{-(y+1)}, y>-1$$ Let $Y_1,..,Y_4$ be a random sample from the distribution defined by the density function above. Let $Y_{(1)},..,Y_{(4)}$ be the ...
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1answer
34 views

Calculate the probability that out of $5$ randomly chosen claims $3$ are of the size $£5,000$

A very crude model for the distribution of claim size, $X$, in a particular situation represents $X$ as a discrete random variable, which takes the values $£5,000, £10,000,$ and $£20,000$ with ...