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

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1answer
74 views

Probability of event in normal distribution

Let $X$ be a random variable that is normally distributed and $X_1,\ldots,X_n$ be (independet) copies of $X$, then we can estimate this probability by using a simple Monte-Carlo estimator: $p := P (X ...
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1answer
32 views

How to get a Gaussian curve fitting a given range of values?

I was trying to find a way to make a gaussian function out of a range of values: $1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12\ 13\ 14\ 15\ 16$ I want the mean to be the most probable value, $8$ and the ...
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0answers
41 views

Change of variables in calculating the integral of multivariable differential entropy

I know that for one dimensional differential entropy of a density function $p(x)$, one has the following formula by change of variables: $$H(p)=\int ...
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2answers
55 views

Weibull Distribution, what is $R^2$?

Given a Weibull Distribution $f_R$, how do I transform $R\to R^2$, and what is the distribution for $R^2$? Attempt: Since $f_R$ is distributed with parameter $k$ and $h$ as a function of $x$, so . ...
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0answers
160 views

Finding the nth moment of the geometric distribution: Why does interchanging the derivative and summation operators not work after n=1?

I am trying (and failing) to find a recursive formula for the $nth$ moment of a geometric distribution. I have arrived at bogus results, and I think it has something to do with the convergence of the ...
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41 views

How to find the CDF of distance between two point in two circles respectively?

Let $C_1$ and $C_2$ be two circles with radius $R$ and $r$, $H$ be the distance between two centers, $H\in[0,+\infty)$, pick up Point $P_1(x,y)$ from $C_1$ and Point $P_2(a,b)$ from $C_2$ uniformly, I ...
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1answer
20 views

Random variable with pdf proportional to Normal

I don't understand the step highlighted in green. I know $f_Z(z)= \frac{k}{\sqrt{2\pi}}$ $ e^{-\frac{z^2}{2}}$ when $z>-\frac{\mu}{\sigma}$ and $0$ elsewhere; but i'm stuck at this point.
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79 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. ...
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1answer
88 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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0answers
36 views

Urgent Find the CDF of U [duplicate]

I am having problems with Part 2. I know the upper limit of Y is x+u in the formula. But what about the limits of X ? Please help me !!
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0answers
46 views

pdf of area of a circle

$X,Y$ are random variables with standard normal distribution (they are independent). $W$ is the area of the circle that has center at $(0,0)$ and passes through $(X,Y)$. What is the pdf of $W$? I ...
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1answer
112 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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2answers
115 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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1answer
34 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
15 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
35 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
11 views

Distribution of sorted sum of elements of random vector

Suppose $x$ is a d-dimensional standard Gaussian vector (zero mean, identity covariance matrix) . What is the distribution of $\sum_{i =1}^m | x_i |$ where $x_i$ is the $i^{th}$ largest entry of $x$ ...
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1answer
78 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
34 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
57 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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2answers
86 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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2answers
34 views

Probability with a uniform distribution

A group of athletes have pulse rates uniformly distributed between 60 and 75. What is the probability that a randomly chosen member of the group has a pulse rate greater than 70? I am thinking that ...
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1answer
62 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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53 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
54 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
44 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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0answers
29 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
42 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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2answers
51 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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1answer
189 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
22 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 ...
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1answer
127 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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3answers
41 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
20 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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1answer
14 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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2answers
361 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
18 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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1answer
48 views

Hypergeometric Distribution Confusion

I'm having trouble understand the part of the pmf for the Hypergeometric Distribution highlighted in green: $$\Pr[X = k] = \frac{\dbinom{m}{k}\!\!\color{green}{\dbinom{N-m}{n-k}}}{\dbinom{N}{n}}$$ ...
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2answers
102 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 ...
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2answers
43 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 ...
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1answer
78 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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1answer
168 views

If $ X = \sqrt{Y_{1} Y_{2}} $, then find a multiple of $ X $ that is an unbiased estimator for $ \theta $.

Problem: Suppose that $ (Y_{1},Y_{2},Y_{3},Y_{4}) $ denotes a random sample of size $ 4 $ from a population with an exponential distribution whose probability density function $ f $ is given by ...
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1answer
41 views

Rescaling function for probability of $k$ adjacent ones in a binary string

Call $\xi$ a random variable taking values in $\{0, 1\}^{\{0, 1, 2, \ldots, n\}}$, where each character of the string has vaalue $1$ with probability $p$ and $0$ with probability $1-p$ independently. ...
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2answers
47 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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0answers
30 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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1answer
26 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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1answer
200 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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0answers
42 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
323 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
25 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 ...