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

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29 views

a 12-side dice with a lot of players probability problem

One million players participate in a game that has 10 levels. At first, all players are at level 1. At the end of each turn, each player rolls a twelve-sided die, numbered 1 to 12. Player advance a ...
-1
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1answer
30 views

How to see (in)dependence of random variables based on their joint density

This is a valid joint pdf. I just want to know if X1 and X2 are dependent or independent rvs ? Why ? Thank you for your help. Is there a way of seeing this without computing the marginal density ...
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1answer
28 views

Exponential Distribution as a density function

I have an important presentation on tuesday about the exponential distribuion as a density function. My question is: What are the advantages of using this function? In order to fulfill my task i have ...
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2answers
23 views

expectation of random variable

I have this question : What is the expected value of $E(X^{100})$ if X is a random variable such that $E(X)=E(X^2)=1$? I am very confused as $X$ could be a poisson or gamma variate.
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1answer
23 views

Independence of two random variables

I am learning probability theory by myself and have a problem with independence of two random variables: Suppose that I have $X_1, X_2, Y_1, Y_2$ are i.i.d N(0,1), then I define: $X=X_1+X_2, ...
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0answers
25 views

Upper bound or approximate form for the CDF of Hypo exponential random variable

The CDF of hypo exponential random variable (sum of $n$ independent exponential random variables $X_{i} $ with different rates $\lambda_{i}$) is given by I seek for an upper bound or an approximate ...
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0answers
14 views

Mixture Models: finding the marginal distibution

I don't understand the part underlined in green for b). $f\left( x|Y=y \right)=ye^{-yx}$ but for this to be positive x does not have to be greater than zero?
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1answer
32 views

Probability: boxes and Discrete probability distribution

1) We have n white balls and 2n black balls. In how many ways we could put it into n boxes if in every box it have to be at least one black ball have to be in every box. My answer is: First of all ...
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0answers
16 views

Transition matrix in left-right hidden semi-Markov model

I'm developing a hidden semi-Markov model left-right . In a left-right model a sequence of $M$ states starts in state $1$ and ends in state $M$, with no repetition of states. Since the model is ...
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1answer
40 views

Realisation of a Poisson process

Is anyone able to explain the section highlighted in green?
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1answer
28 views

number of ones with neighbours in a random binary string

Consider a sequence of i.i.d. random variables $(\xi_i)_{1 \leq i \leq L}$ such that $\xi_1 \in \{0,1\}$ and $P(\xi=1)=p$. Introduce the function $N : \{0,1\}^{L} \rightarrow \mathbb{N}$ which counts ...
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0answers
16 views

A variation of Polya's urn

Polya's urn model is as follows: you have $a\in \mathbb{Z}_{>0}$ red balls and $b\in \mathbb{Z}_{>0}$ blue balls in a urn. Suppose you pick a red ball. Then you put back $c\in ...
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2answers
19 views

Question about independence

First of all is true that given $X,Y$ two random variables indenpendent; $(X,Y)\in D\subset \mathbb{R}^2$ then $\text{Cov}(X,Y)=0$? I tried to prove it and this is my solution: If $D=[a,b]\times ...
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1answer
29 views

Find $k$ if $f(x)=k(1/2)^x$ for $x=1,2,3,$ and $0$ elsewhere.

Find $k$ if $f(x)=k(1/2)^x$ for $x=1,2,3,$ and $0$ elsewhere. Here is what I did: $$\int_1^3 k\left(\frac12\right)^{x}dx=1$$ After integrating, I found it to be: ...
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0answers
38 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 ask this ...
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0answers
36 views

Pdf of this estimator

We have a set of unidimensional data, $X_1, \ldots , X_n$ drawn from the positive reals. We define a model for its distribution: The data are drawn from a uniform distribution on the interval $[a, ...
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1answer
89 views

Can someone please help to understand the following probability

I was reading something on communication, then I came across the following equation: $Power_{rx}=Power_{tx}*|R|^2/(1+d^2)$ where $Power_{tx}$ and $d$ can be assume to be constant, and R is the ...
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2answers
35 views

Probability Generating Functions with Three Dice

Three identical dice are thrown. The dice are fair, that is, for all three dice the probability of turning up face $j$ is $1/6$, $1 \le j \le 6$. Let $X_1,\ X_2,\ X_3$ be the independent random ...
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2answers
46 views

Approximate distribution for the sample mean?

A random variable $X$ is said to follow a discrete uniform distribution if its probability function is given by $$p_X(x) = \left\{ \begin{array}{ll}\frac{1}{\theta}, & x = 1, 2, \ldots, ...
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2answers
37 views

Uncorrelated, Non Independent Random variables

I don't understand the parts highlighted in green. I understand that the supports imply that X and Y are not independent but not how the graph shows this graph. I'm a bit confused by all aspects of ...
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1answer
46 views

The Weibull as the limiting distribution of the Burr distribution

I often deal with "payout patterns" which are vectors of the cumulative percentage of a loss that has been paid over time. For example, for $t \in [0, 1, 2, 3, 4, 5]$ I may have $p_t = (5\%, 15\%, ...
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0answers
8 views

Using the simplex to define a probability distribution

I am reading a paper where a probability distribution with n categories is being defined in terms of a vector $\mu = <m_1, m_2, ... m_n>$. The authors define this as a vector $\mu \in ...
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1answer
48 views

How to draw a distribution of the function?

I have $f(x)= \frac{1}{2}$ and $x\in[0,2]$. How i can draw a distribution for it ?
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1answer
17 views

Getting the pmf from probability generating function?

I uploaded a picture so that my question may be accurate. It is question 3a that I am struggling with. I've learnt that the P(X=r)is the rth derivative(w.r.t. t) of the pgf at t=0 . Divided by r ...
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1answer
35 views

Bivariate Transformation

Why can I not let $V=X$ in this transformation as opposed to $V=Y$? I have tried it with $V=X$ and i get a different joint pdf.
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2answers
82 views

Sum of two truncated gaussian

What is the CDF and the PDF (or approximation) of the sum of two truncated gaussian $X = TN_x(\mu_x,\sigma_x;a_x,b_x)$ and $Y = TN_y(\mu_y,\sigma_y;a_y,b_y)$ ? where $TN(\mu,\sigma;a,b)$ is a ...
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0answers
29 views

Expectation and Variation of dependent RVs

This is a really nice question, and while I can think of a solution to both parts, I wonder if there's a more elegant one to the latter: A fair 6-faced dice is tossed once. In a box there are 6 ...
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2answers
34 views

Prove the existence of exactly two maxima for a positive $L^1$ function

I have a function $f:\mathbb R \rightarrow \mathbb R^+$ with the following properties it is $L^1$ and $C^2$ it has one single extremum (maximum) at $x=0$ it is symmetric: $f(x)=f(-x)$. it is ...
0
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1answer
65 views

Finding expected value of mean of an estimator

We have a set of unidimensional data, $X_1, . . . , X_n$. : The data are drawn from a uniform distribution on the interval $[a, b]$. This model has two positive real parameters, a and b, such that $0 ...
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1answer
40 views

probability in roulette!

So I have read how to play roulette...still a little confused, and now I'm faced with a probability question about it which makes the problem a little harder. Please help me reason this where ...
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2answers
24 views

A joint PDF question: $\displaystyle f(x,y)=1-\frac x3-\frac y3$

Really stuck on this problem: If $\displaystyle f(x,y)=1-\frac x3-\frac y3$ for $0 \le x \le 2$ and $0 \le y \le h,$ the find $h$. I know I need to integrate but confused how to set it up. ...
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1answer
59 views

Finding the pdf of an estimator

We have a set of unidimensional data, $X_1, \ldots , X_n$ drawn from the positive reals. We define a model for its distribution: The data are drawn from a uniform distribution on the interval $[0, ...
1
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1answer
60 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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2answers
42 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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1answer
35 views

How does a pdf of the difference of two random variables relate to the pdf of each random variable

Let $T_1$ and $T_2$ be non-negative continuous random variables (rv) denoted in the form $T_i = \mu_i + \sigma_i X_i$ for $i=1,2$ where \begin{eqnarray*} T_{1} &=&\mu _{1}+\sigma _{1}X_{1} \\ ...
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0answers
37 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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1answer
23 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
10 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
16 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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0answers
40 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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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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1answer
58 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
28 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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0answers
24 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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1answer
11 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
27 views

Marginal PDF with dependent variables

I don't understand how to work out the limits of integration in b).
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0answers
7 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
29 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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34 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
56 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 ...