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

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14 views

Distribution and expecation value of ceiling function of poisson

There is poisson random variable $X$ $$P(X=x)=\frac{\lambda^{x}}{x!}e^{-\lambda}$$ And define random variable $Z=\lceil \beta X \rceil$ ( $\beta$ is rational number which is lower than 1). How can i ...
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0answers
7 views

Failure boundary for simple routing problem

As an absolute beginner concerning probability theory I am currently trying to solve the following problem: Given a grid that has $x$ columns (here $x = 4$) and $y$ rows (here $y = 5$), we insert a ...
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0answers
10 views

Continuity of random variable as function of a random variable

Suppose, we are given a continuos random variable $X$ and a continuous and nondecreasing function $f$. Can it be shown that a second random variable $Y=f(X)$ is continuos on the support of $X$? What ...
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1answer
19 views

Ehrenfest Chain: stationary distribution

In the Ehrenfest Chain model: There are M balls which are divided between urn A and urn B. At each stage, if a ball is chosen, then it would be moved into a different urn. Let $X_n$ be the # of ...
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0answers
8 views

Expected values of Hermite polynomials under Gaussian distribution

On Wikipedia there's a nice result stating that $$E[He_n(X)]=\mu^n,$$ where $He_n$ is the (probabilists') Hermite polynomial of order $n$ and $X$ is a $N(\mu, 1)$ random variable. I'm interested in ...
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2answers
15 views

Find PDF on $[0,6]$ such that $P([1,3]) = 0.5$

Find a probability density function $f$ on $[0,6] \subset \mathbb{R}$, such that $\mathbb{P}([1,3]) = 0.5$ That is we need to find an $f$, such that $\int_{[0,6]} f(x)dx = 1$ and $\int_{1}^{3} ...
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0answers
9 views

Continuity of Monte-Carlo simulations with uniformly distributed input parameters

Suppose a continuous and monotone function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ to be given. So, in the general case, if I slightly change parameters $a$ and $b$, the function ...
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21 views

What is expected value of only positive numbers [duplicate]

Normal distribution with mean zero and standard deviation 1. What is the expected value of just the positive variables
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2answers
29 views

Distribution of the sum of $N$ loaded dice rolls

I would like to calculate the probability distribution of the sum of all the faces of $N$ dice rolls. The face probabilities ${p_i}$ are know, but are not $1 \over 6$. I have found answers for the ...
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1answer
14 views

Method for determining distributions of sum of Normal distribution unknown mean and variance

I've been trying to complete this question but have been struggling to see how to approach it. Any help would be greatly appreciated. Is there a standard way of approaching and answering ...
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3answers
47 views

Upper bound for difference of Poisson random variables

Let $X, Y$ be random variables with Poisson$(\lambda)$ and Poisson$(2\lambda)$ distributions, respectively.Then (i) If we assume that $X, Y$ are independent, $$\mathbb{P}(X \geq Y) \leq ...
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2answers
35 views

Methods for calculating the mean and variance of a distribution created from the addition of two normally distributed quantities

I'm trying to understand how to interpret the following which refers to determination of the mean and variance of a distribution that's the result of adding two normally distributed random variables. ...
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0answers
12 views

Position error probability distribution when distance and angle error distributions are zero mean Gaussian

In one problem we are estimating the position of an object from the measurement of its distance $\mathbf{r}$ from a point as well as its angle $\mathbf{\theta}$ from the reference direction. The error ...
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0answers
11 views

Two-alternative forced choice

Suppose that $p[r|+]$ and $p[r|-]$ are both Gaussian functions with means $\langle r \rangle_+$ and $\langle r \rangle_-$ and common variance $\sigma_r^2$. How can I show that $$P[correct] = ...
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0answers
23 views

Proportional probability of payouts with defined expected value.

Assume we have a lottery with payouts like this $(2,3,5)$ So you buy a ticket and you can win a pot which will multiply your ticket price by the numbers written ahead.The organizer expects a margin ...
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2answers
14 views

Estimate on Probability of a standard normal variable

In the book written by Karatzas & Shreve, at the page - 111; the authors have mentioned about a result: If $Z_{v}$ be a standard normal variable; then for $\epsilon \gt 0$ ; $\mathbb P ...
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1answer
16 views

Show that Uniform$(1,5)$ is neither singular nor absolutely continuous with respect to Uniform$(0,3)$.

Actually, I'm just studying singular continuity, absolute continuity.I know the definitions.And have solved few very basic sums. Now, in this problem, I'm not understanding what does this 'with ...
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0answers
8 views

Statistical distance between a multiplicative mask and a random number

Given $x \in \{1,\ldots,2^n\}$ and a uniform random $r \in \{1,\ldots,2^{n+k}\}$, then the statistical distance $\Delta(x + r\bmod q; r) < 2^{-k}$, for a $q > 2^{n+k+1}$. With addition this is ...
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2answers
50 views

How to integrate $\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$ where $a>0$

How to integrate $$\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$$ where $a>0$ The real problem is this integral $$\lim\limits_{\alpha\rightarrow 2}\int\limits_0^\infty e^{-a x^\alpha}\cos(b x) ...
1
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1answer
9 views

Expectation of the minimum of two $\mathcal U(0, 1)$ r.v.'s conditional on it being greater than or equal to some value

Let $X_1, X_2$ be i.i.d. $\mathcal U(0, 1)$ (continuous) r.v.'s, and let $0 \le R \le 1$ be some number. What is $\mathbb E[\min(X_1, X_2) \mid \min(X_1, X_2) \ge R]$? My attempt: Let $Y = ...
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2answers
27 views

Convergence of Sum of Random variable to another - Cantor function

Let $(X_{n})_{n\geq1}$ be i.i.d. Ber$\left(\frac{1}{2}\right)$. I want to show that $$\sum_{{n\geq1}}\frac{2X_{n}}{3^{n}}$$ converges almost surely to a random variable $X$, without saying that this ...
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0answers
9 views

Sampling Distributions. Statistics [on hold]

I'm stuck in this problem: Problem Picture I did the literal a and b, but the rest of them I don´t understand, the reason why.
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0answers
31 views

On the probability distribution of iterated permutations

I have this little problem that has been nagging me for a couple of months now. It occurred to me when considering the fairness of card shuffling methods. Here's my best attempt at formalizing it: ...
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2answers
32 views

What probability mass, density or distribution function might corresponds to this moment generating function? [duplicate]

I have somehow come up with a random variable $X$ with moment generating function (assuming it exists) $$M_{X}(t) = -t (1 - e^t)$$ What is the probability mass, density or distribution function? It ...
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0answers
46 views

Expectation and Variance of $X/(X+Y+Z)$

I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal ...
2
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1answer
20 views

Limiting Distribution $\Delta-$method

Let $Y_n\sim \chi^2(n)$. What is the limiting distribution of $U_n= \dfrac{\sqrt{Y_n}-\sqrt{n}}{\sqrt{2}}?$. What I know is that if $X_i\sim \chi^2(1)$, I can write $Y_n = \sum\limits_{i=1}^n X_i$. ...
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0answers
28 views

Calculate the discrete density of the variables of a Markov chain

$X$ and $Y$ are independent random variables of Bernoulli with parameter $\frac{2}{3}$. $Z=X+Y$ $\{X_n\}_{n \in \mathbb{N}}$ with values in {0,1,2} having $Z$ such as initial law and the transition ...
2
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1answer
14 views

Inverse of Gaussian CDF, Sum

Consider the following setting. Let $k = 1, \ldots, n$ and define $$y_k= \Phi^{-1}\left(\frac{k}{n+1}\right),$$ where $\Phi$ is the inverse of the CDF of a standard normal. I noticed numerically ...
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1answer
14 views

What is support of sum of two random variable

Suppose there is random variable $K= X + Z$ $$P(X=x)=\frac{T\lambda^{x/\alpha}}{(k/\alpha)!}e^{-T\lambda}\quad(x=0,\alpha,2\alpha,......)$$ ...
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1answer
39 views

How can i find distribution of ceiling poisson

There is poisson random variable $X$ $$P(X=x)=\frac{\lambda^{x}}{x!}e^{-\lambda}$$ And define random variable $Z=\lceil \beta X \rceil$ ( $\beta$ is rational number which is lower than 1). How can i ...
0
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0answers
15 views

independence of chi square distributions

We already knew that if two independent chi-squared random variables, then their sum is also chi-squared with the degree of freedoms is the sum of theirs. How about the converse? If $X\sim\chi^2(n)$ ...
0
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1answer
62 views

Distribution of a function of a random variable

Suppose we have continuous random variable $X$ with distribution $f_X$. That is $$ P\left(a \le X \le b \right) = \int_{a}^{b} f_X(x) \ dx $$ Now suppose I have a function $\phi: \Bbb{R} ...
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2answers
28 views

Obtaining probability density function $f_Y(y)$ when we know joint probability distribution $f(x,y) = 1/(x+1)$

Suppose joint probability density function is $f(x,y) = 1/(x+1)$ for $0<x<1$ and $0<y<x+1$. I try to calculate marginal density function $f_Y(y)$ by $$f_Y(y) = \int_{y-1}^1 ...
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1answer
21 views

Finding bivariate probability mass function (by counting?)

Suppose that we role $d$ dice. Let $X, Y$ be random variables, where $X = \#$ rolled by the die with the highest value. $Y = \#$ rolled by the die with the second highest value. By convention, we ...
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19 views
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25 views

Which distribution would be the most appropriate?

What standard distribution would be suitable for the random phenomenon at hand, and what are the knowns and unknowns? e) The size of an automobile insurance claim I'm thinking that the distribution ...
1
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1answer
21 views

Joint distributions where one is uniform

Let $X$ have a uniform distribution on the interval $(0,1)$. a) Find the c.d.f. and p.d.f. of $Y=\dfrac{X}{1-X}$. b) Find the c.d.f. and p.d.f. of $W=\ln Y$. I am extremely confused on ...
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0answers
24 views

A functional of a Lévy process

Does anyone know if there are any papers/results on functionals of the type : $$\int_0^tp(X_s)ds$$ where $X$ is a Lévy process and $p$ is a polynomial. For example, is the distribution of such an ...
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0answers
58 views

Hypergeometric distribution with a priori probabilities of the balls

If we have a urn with $N$ balls of two colours ($D$ red and $N-D$ black balls respectively), then probability of having $k$ red out of $n$ balls drawn at once without replacement follows the ...
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1answer
18 views

Convergence in distribution - X/Y

If X(n) and Y(n) converges to X and Y in distribution respectively then does X/Y(n) also converge to X/Y in distribution? Prove or disprove. I feel that this is correct but have not been able to ...
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1answer
30 views

There is problem in calculating pgf(probability generating function)

I posted question about distribution of poisson distribution multiplied by constant. Here! From this post, i can obtain what i want. $$P(X=x)=\frac{\lambda^{n}}{n!}e^{-\lambda}$$ $$Z=\alpha X ...
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1answer
20 views

Difficulty in finding marginal distribution

Let $X=(X_{1},X_{2})$ have joint pdf.$$f(x_{1},x_{2})=\begin{cases}\frac{e^{-\frac{x_{2}^2}{2}}}{x_{2}\sqrt{2\pi}},\ &\text{if}\ 0<|x_{1}|\le x_{2}<\infty.\\0,\ &\text{otherwise} ...
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1answer
27 views

What is distribution of Poisson multiplied by positive constant

Let $X$ is poisson distribution. $$f_{X}(n;\lambda)=\frac{\lambda^{n}}{n!}e^{-\lambda}$$ And there is some positive constant $\alpha$. I like to know pmf(probability mass function) of $Z=\alpha X$. ...
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1answer
38 views

Two random variables X and Y follow the same distribution. Then

Two random variables $X$ and $Y$ follow the same distribution. Then The distribution of $X − Y$ must be symmetric about $0$. The median of $X − Y$ must be zero. The median of $X + Y$ is twice of ...
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0answers
20 views

Proof of the DKW inequality

My goal is to prove the following inequality, known as the Dvoretsky-Kiefer-Wolfowitz inequality (1956) : Let $(X_i)_{i \geqslant}$ be iid random variables. Let $\displaystyle F_n(x)= ...
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1answer
25 views

Calculate the estimators of $E[X]$ and $Var[X]$ using the method of moments

$(X_1,\dots, X_n)$ is a random sample extracted from a uniform distribution on the interval $$(\alpha-\beta, \alpha+\beta) \ \ \ \ \alpha \in \mathbb{R}, \beta \in \mathbb{R}^{+}$$ Demonstrate ...
2
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1answer
7 views

Clarification on the Poisson distribution

Given that an event occurs at some rate $\lambda$ per unit time, I know that the probability that n such events occur in unit time is given by the Poisson distribution. $P(n) = ...
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32 views

Is the Covariance of Two Random Variables Convex or Concave or Neither?

Are there any standard results established regarding the behavior of the Covariance of two random variables? For example, whether it is a convex or concave functions and so on and under what ...
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39 views

Proof of inequality 5 [closed]

Let $B \subset F$ be a sub $\sigma$-algebra, inequality $|x||y|/\alpha\beta \leq x^2/2\alpha^2+y^2/2\beta^2$ with $\alpha\beta>0$ to prove $E\{|xy| \mid B\} \leq \alpha\beta$.
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16 views

Convergence in distribution (sufficient conditions)

Define a set of random variables $Z^2_i=\frac{(v_j-np_j)^2}{np_j} $ $K(Z_i,Z_j)=-\sqrt {p_i p_j}$ $E( Z^2_i ) = 1-p_i$ and $T_i= g_i - \sum_{j=1}^r g_j \sqrt p_j \sqrt p_l$ $K(T_i,T_j)=-\sqrt ...