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

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calculating probabilty of variable belonging to a probabilistic range

Suppose there are two random continuous variables $x$ and $y$ always non-negative, i.e. they do not assume negative values at all. They have their density functions given, $p_x$ and $p_y$. How do i ...
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1answer
13 views

Re-writing exponent of Multivariate Gaussian

In Bishop's Pattern Recognition and Machine Learning (ISBN-13: 978-0387-31073-2), Bishop writes on page 86: This is an example of a rather common operation associated with Gaussian ...
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2answers
53 views

PDF of $Z=\frac{X^2+Y^2}{2}$ where $X\sim N(0,1)$ and $Y\sim N(0,1)$

Say $X \sim N(0,1)$ and $Y\sim N(0,1)$ are independent random variables. So: $f_X(x) = \frac{1}{\sqrt{2\pi}}e^{\frac{-1}{2}x^2}$ and $f_Y(y) = \frac{1}{\sqrt{2\pi}}e^{\frac{-1}{2}y^2}$. Now I am ...
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1answer
21 views

$n$ dice, finding $\operatorname{var}(X)$

If we throw $n$ dice. And $X$ is the total number of eyes. Find $\operatorname{var}(X)$. My idea was to label $X=X_1+\cdots+X_n$ where $X_1$ is the outcome of die $1$ etc. And because $X_1,\ldots, ...
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1answer
34 views

Distribution of $1/(1+X^2)$ if $X$ is standard Cauchy

Let $X$ be a Cauchy random variable with parameter $1$ i.e. with density $\dfrac{1}{\pi(1+x^2)}$. What is the density function of $Z:=\dfrac{1}{1+X^2}$? My attempt: Say $\phi(x) = ...
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1answer
25 views

Application Problem of Expected Value of Posterior Distribution

I am trying to understand the following: Suppose that the number of people who visit the grocery store on any given day is Poisson($\lambda$) and the parameter of the Poisson distributed has a ...
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2answers
36 views

Writing the pdf for a Gamma Distribution

Let $X_1, \dots, X_5$ be 5 independent variables from the exponential distribution with the mean $2$. a) The pdf of $T=X_1 + \dots + X_5$ b) $P(T > 5)$ c) $E(T/5)$ Having a little trouble ...
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38 views

How many elements use for device to work with certain probability.

Hello I have exercise like this: Some device is build from $n$ elements. Device works if at least 97% of elements is working. Probability of each element breaking is $0.02$ . How many elements must ...
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2answers
23 views

Determining what distribution a given random variable has.

Let $X$ be a poissonian random variable with an expected value of 100. (I guess it means $\lambda=100$). Let there be a kindergarten with $X$ kids. On a certain day, the teacher decides to split ...
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17 views

Multiplying a non-central chi square distributed variable by a constant value

Let $X \sim \chi_k^2(\lambda)$, that is a non-central chi square distributed random variable with $k$ degrees of freedom and non-centrality parameter $\lambda$. What is the distribution of $Y=cX$, ...
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47 views

Almost sure convergence, product of variables

Consider the sequence of random variable given by distribution : $$\mathbb{P}(X_n=1)=1-\frac{1}{n},$$ $$\mathbb{P}(X_{n}=0)=\frac{1}{n}$$ and $Y_n=X_n \cdot Y$ for random variable Y. Does the $Y_{n}$ ...
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1answer
68 views

Find the distribution of linear combination of independent random variables

Given independent and identically distributed random variables $X_1, X_2, \dots, X_n$, each of them has the same p.d.f $f(x) = Pr(X = x)$ on support $(a, b)$. How do I find the pdf or cdf of $Y = ...
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0answers
41 views

What is the mean of this random variable?

Define the following random variable $X_{n, \delta}$, for integer $n$ and $0 < \delta < 1$: I have an urn with $n$ different balls in it. I sample $n$ balls (with replacement), and I keep a ...
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0answers
17 views

Sum of two gamma-distributed random variables

Let $X_1 \sim \Gamma(\alpha,\beta_1)$ and $X_2 \sim \Gamma(\alpha,\beta_2)$, what is the distribution of $Y = X_1+X_2$? Can it be expressed in terms of a certain probability distribution? Thanks
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1answer
33 views

Find the mean and variance of $V_n=\frac{1}{n}\sum_{i=1}^n(X_i-u)^2$

Suppose that $X_1,X_2,...,X_n$ is a random sample from a distribution with mean $\mu$ and variance $\sigma^2$. Suppose also that $v:=\mathbb{E}[(X_1-\mu)^4]<\infty$. Find the mean and variance of ...
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0answers
11 views

Expectation of an absolute exponential generalized beta 2(EGB2) distributed variable

For a bachelor thesis i'm trying to impose a new GARCH model using an non-linear exponentioal GARCH model with an underlying Generalized beta distribution of the second kind(introduced by ...
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1answer
19 views

Consider joint p.d.f. $f_{X,Y}(x,y)=C_1e^{-x-y}$, where $0<x<y<\infty$. Find $C_1$

Consider joint p.d.f. $f_{X,Y}(x,y)=C_1e^{-x-y}$, where $0<x<y<\infty$ with continuous random variables $X$ and $Y$. Find $C_1$. Not sure how to approach this. The main property that I am ...
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1answer
43 views

Posterior Distribution and Expected Value of a Coin Toss where Probability of Heads is a Random Variable

I am trying to solve the following: Suppose X is the number of times a coin is tossed until a heads is observed. Let Y denoted the probability of observing heads and assume $f_Y(y)=ky^2$, ie the ...
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28 views

Does this Expected Value not exist?

Let $f_{X\mid Y}\sim F_s(y)$, the first success distribution, and $f_Y=ky^2$. I want to compute $E[X]$. Using the fact that $E[X]=E_{y}[E_{x}[X\mid Y]]$. $E_{x}[X\mid Y]]= E[F_{s}(y)]=1/y$. Thus, ...
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1answer
32 views

Sum of weighted chi square distributions

Let $X_1 \sim \chi_{k}^2$ and $X_2 \sim \chi_{k}^2$ (both i.i.d) and $a_1$ and $a_2$ positive real values. How can be expressed the PDF of $Y = a_1X_1 + a_2X_2$? Is it also a chi-square distribution? ...
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Why the marginal distribution function F_x(x) = F_xy(x,∞)?

I don't know if the next reasoning is all right. If the joint distribution function $F_{xy} = (\infty,\infty)$ = $ P(-\infty<X<\infty,-\infty<Y<\infty)$. That's the probability of the ...
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1answer
41 views

Data that follows a distribution with non-finite expectation

I find it difficult to get around the idea of some random variables following some distributions (such as the Cauchy Distribution) not to have finite means. How does one actually interpret data from ...
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1answer
25 views

Notation for the limit to the boundary of the support of a function

So let $f(z)$ by a density function for $z\in\operatorname{supp}f$. In some cases (for example when $f$ is the pdf of the normal distribution), $\operatorname{supp}f$ will be $\mathbb{R}$ and one ...
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2answers
43 views

Integral involving the CDF of normal distribution bis [closed]

I got stuck in computing this integral, and I must say your help will be very much appreciated! Here is the integral: $$\int_{-\infty}^y x*e^{bx+c}*N(\delta x+\varepsilon) dx $$ Where $y \in ...
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25 views

what does it mean for CDF distribution to be unbounded? Example?

What does it mean for cumulative distribution function (CDF) to be unbounded? What would be an example distribution?
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2answers
57 views

Infinite intersection of an interval and probability of selecting a random point

I am attempting to solve the following: Let $A_n= (\frac{1}{2}-\frac{1}{2n}, \frac{1}{2}+\frac{1}{2n})$. Show that $\bigcap\limits_{n=1}^\infty A_n=\{\frac{1}{2}\}$. Then apply the continuity ...
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0answers
15 views

How to determine the pdf for a model in phase space representation?

Consider a univariate discrete linear model : $z(k) = y(k) -(a* z(k-1) + b * z(k-2))$ where $y(k) = x(k) + \eta(k)$ $x(k) = s(k) + p*s(k-1) + q*s(k-2)$ is a Moving Average model of order 2. ...
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9 views

Distribution of the maximum vector (over i.i.d. set) and its dot product with the eigenvectors spanning the rest. [migrated]

Let $z_1,\dots,z_n$ be i.i.d. draws from $N(0,\Sigma)$, where $\Sigma$ is a $p\times p$ matrix. Assume that $p>n$. Suppose (up to re-labeling) that $z_n=\max_i \|z_i\|_2$. Consider the eigenvectors ...
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2answers
49 views

Probability of a random matrix to be invertable.

Suppose that $x_{ij},i=1,2,\ldots,n;\,j=1,2,\ldots,m$ are independent and identically distributed continuous random variables. What is the probability that the group of vectors ...
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33 views

Calculating the density function $f_X$ from joint density $f_{X,Y}$

X and Y have continous distribution, the joint distribution is $f_{X,Y}(x,y)=\frac{1}{2\pi\sqrt[]{1-p^{2}}}e^{-\frac{1}{2(1-p^2)}(x^2+y^2-2pxy)}$, ($p$ is a constante) We need to find the ...
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2answers
21 views

Inverse cdf function of monotone transformations of random variable

Suppose $X\sim\mathcal{F}_X$ and denote $F_X(x)$ the cdf and $F_X^{-1}(x)$ the quantile function of $\mathcal{F}_X$ evaluated at $x$. Now define: $Y=\exp(X)$ and denote $\mathcal{F}_Y$ the ...
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0answers
25 views

Prove or disprove that the Bhattacharyya distance is a true distance function

Let $\mathcal{X}\equiv\Bbb{R}^n\times\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ is the space of all symmetric positive-definite $n\times n$ real matrices. Let $x,y\in\mathcal{X}$, where $$ ...
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0answers
29 views

Gaussian Distribution Under Orthogonal Transformation

Let $\mathbf{H}\in \mathbb{R}^{n\times n}$ be a random matrix whose every element has a Gaussian distribution with mean $m_{ij}$ and variance $\sigma^2$ i.e. ...
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1answer
16 views

Find the distribution of some random variable connected to Wiener Process. Please, check my solution.

I need to find a distribution of $ 5W_1-W_3+W_7 $, where $W_t$ stands for Wiener Process $W_t\sim\mathcal{N}(0,t)$. Is this solution right? $E(5W_1-W_3+W_7)=5E(W_1)-E(W_3)+E(W_7)=0$ and since ...
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1answer
48 views

Will someone please explain multivariate normal distributions with a real-life example?

I understand a concept best when I see it being applied in the real world.
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33 views

Bernouli Trial Probability of Stopping After X Trials

The probability of a trial being a success if 0.30 Trials are repeated until 6 are successful. I'm asked to find the probability that the trials are ended after the 7th. (The 6 successful trials ...
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3answers
59 views

Probability of not choosing from a lot [closed]

Say I have a lot of $1000$ televisions. $10$ out of every $1000$ are defective. When I choose $30$ televisions to test them, how would I calculate the chances of not choosing a defective one from ...
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1answer
32 views

Find the $p_{Y|X}(y|x)$ without the jointly probability

Let the distribution $Y = X + N$. Where $X$ and $N$ are independents and they have distinct distributions. I have $f_X(x)$ but I don't have the $f_{XY}(x,y)$ to use, for example, the following ...
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47 views

Determining the Expected value of a random variable

Suppose we have a Poisson process of parameter $\lambda$. Each event of this Poisson process represents a start date of a period which duration is a random variable that follows an exponential ...
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1answer
30 views

Conservation of Kinetic Energy in Vlasov-Poisson System

I'm studying the very basics of kinetic theory in Vlasov Poisson Systems, and the first equation I'm studying is the free transport equation, i.e.: $$\frac{\partial f}{\partial ...
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14 views

Show $\int_{-\infty}^{\infty}\,f(u,t)dG(u)$ is a ch.f. where $G$ is a d.f. ; $f(u,\cdot)$ is a ch.f. and $f(\cdot,t)$ is continuous.

Show $$\int_{-\infty}^{\infty}\,f(u,t)dG(u)$$ is a ch.f. where $G$ is a d.f. ; and $f(u,\cdot)$ is a ch.f. for each $u$ and $f(\cdot,t)$is coutinuous for each $t$. Note that ch.f. means ...
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1answer
14 views

Finding a CDF given a PDF using summations

I am in a prob and stats class and we have just begun our discussion on discrete random variables. I am given a pdf of $$ f(x) = \left\{\begin{aligned} &x/10 &&: x = 1,2,\ldots,4\\ ...
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0answers
34 views

Poisson distribution of a sum.

Suppose the number of robberies of a clothing store in a random day is a random variable with Poisson distribution with $\lambda=5$. $X_i$ is the number of robberies in day $i$. ...
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1answer
29 views

Find the probability of selecting exactly $14$ defective items.

$70\%$ of items are defective. You randomly select $20$ items. Find the probability that the number of defective items is exactly $14$. I have $n$ as $20$, $x$ as $14$, $p$ as $.7$ and $q$ as ...
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1answer
28 views

Determine the expectation E(XY) of Joint PDF

I am practising some exam questions and am failing to understand the problem at hand. I believe I am supposed to take the double integral of the joint PDF that can be calculated by noting that ...
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0answers
15 views

Joining heterogeneous, discrete probability mass functions

Suppose we have a collection of discrete probability mass functions with different ranges, all of which are from 0 to some positive integer. As a simple example, we might be rolling 3 6-sided dice, 1 ...
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1answer
30 views

What's the probability of obtaining exactly 3 C's out of 10 exams?

The result of an exam consists in three possible grades: A, B and C, each with equal probabilities. What's the probability of obtaining exactly three C's out of 10 exams? And what's the probability ...
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1answer
26 views

Do the set of all standardized moments of a dataset completely and uniquely define it?

I have two datasets, 'A' and 'B', comprising N measurements of one quantity, that I would like to compare to the results of a simulation, let's call this last dataset 'S'. This comparison got me ...
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1answer
93 views

A question in combinatorics

Given a sequence of $0$s and $1$s think of it as blocks of $0$s and $1$s. Like $0001101001$ is a sequence of blocks $000$,$11$,$0$,$1$,$00$,$1$ How may ways can one pick $t$ bits from a $0/1$ ...
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1answer
55 views

Is this PMF or PDF?

I am reading a technical report on expectation-maximization (EM) algorithm (http://melodi.ee.washington.edu/people/bilmes/mypapers/em.pdf) and I am confused about something. For HMMs, it defines ...