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

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15 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
21 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
14 views

Distribution of ceiling function of poisson

I posted question about distribution of ceiling function of poisson. Here! And get some answer but sum of probability is not 1.(To check it i use matlab) So again i post same question. --Question ...
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0answers
41 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
19 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
25 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
12 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
37 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 ...
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0answers
10 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 $X\sim\chi^2(n)$ ...
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1answer
51 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
20 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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0answers
24 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 ...
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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
21 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
56 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
32 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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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
24 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 ...
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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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0answers
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 [on hold]

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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0answers
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 ...
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1answer
12 views

How to cluster words using Jensen-Shannon Divergence?

I'm working on a project that requires text analyzing. I'm currently trying to extract keywords from a single document. I referred to this thesis(Keyword Extraction from a Single Document using Word ...
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2answers
36 views

A priori probability in Bayesian inference problem

The problem A psychic uses a five-card deck to demonstrate ESP, claiming to be able to guess a card correctly with $0.5$ probability (of course, ordinary guessing is $0.2$). A single experiment ...
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1answer
37 views

Finding the Distribution of Y given $X_1 + X_2$ where X, Y ~ Poisson $\Lambda$

So, because this is honestly homework for a course, I'm primarily looking for a hint from where I've gotten so far. The question is very quick. $X, Y$ are independently distributed Poisson ...
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0answers
22 views

Derive a CDF for a random variable based on its relationship with a second random variable? [duplicate]

Take a lifetime with the CDF $F(t)=1-(1-t)^n$ for $t$ in $[0,1]$ and some natural $n$. Now find the CDF of the variable $T_x=T-x$ when $T>x$ for $x$ in $(0,1)$. I need help getting started on this ...
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0answers
33 views

I like to prove sum of new discrete random variable`s pmf is equal 1

First of all from last question I can obtain some discrete random variable which described below. In this page, we denote this random variable as $Z$. $P(Z=0) = ...
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0answers
15 views

p-value of probability distribution

Suppose $\{X_{i}\}_{i=1}^{50}$ are independent and identically distributed samples from the following probability distribution: $$(1/\theta)\exp(-x/\theta); \hspace{1mm} x>0.$$ Given ...
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1answer
8 views

How to calculate the transition density for a multivariate jump process

I have the following stochastic process: $dX = (A-I)XdN$, where $X$ is a $2\times1$ vector of random variables, $A$ is a constant, real, symmetric, $2\times2$ matrix, $I$ is the identity matrix and ...
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1answer
26 views

Expectations and densities

Because of an article that I'm trying to understand, I've come up with the folowing question: Suppose we have $f:(\mathbb{R}_{\geq 0})^2 \rightarrow \mathbb{R}_{\geq 0} $ , $\ X,Y\geq 0 \ \ $ ...
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0answers
20 views

Implications of convergence in distribution

I would like to ask you for an help to show $(\star)$ Consider a sequence of real-valued random variables $\{X_n\}_n$ and assume $\sqrt{n}(X_n-\mu)\rightarrow_d (N,\sigma^2)$ as $n \rightarrow ...
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3answers
43 views

How can I show that $X$ and $Y$ are independent and find the distribution of $Y$?

$X_1,X_2,\dots,X_n$ is an i.i.d. sequence of standard Gaussian random variables. \begin{align}X&=\frac{1}{n}(X_1+X_2+\dots+X_n) \\[0.2cm] Y&=(X_1-X)^2+(X_2-X)^2+\dots+(X_n-X)^2\end{align} ...
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2answers
27 views

Distribution of product of bernoulli random variable and poisson random variable

There are random variable $Z=XY$ ($X$ is poisson and $Y$ is bernoulli) $$X(n;\lambda) = \frac{\lambda^n}{n!}e^{-\lambda}$$ $$Y=\begin{cases} & \beta \text{ with probability } \beta \\ & 0 ...
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1answer
30 views

Any alternate method to find probability of function of two random variables?

Let $X$ and $Y$ be two independent random variables such that $X\sim U(0,2)$ and $Y\sim U(1,3)$. Then $P(X<Y)$ equals I can do this question using transformation $U=X/Y$ ,$V=Y$ as ...
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0answers
21 views

Expected value of a distribution with 2 variables [on hold]

can anyone help me find the expectd value vector and variance matrice of this model. Y1 and Y2 is 0 or positive integers
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0answers
62 views

Proof that Derivative of Expected Value is Zero (Using Differentiation show Unconditional Expectation is Constant)

If the expected value of a distribution is constant, it means its derivative with respect to the values it can take must be zero. I was wondering if there is a rigorous proof of the same. Steps Tried ...
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1answer
32 views

CDF of Sum of a Guassian and Exponential random varaible

If $X$ is a Gaussian random variable with mean $\mu$ and variance $\sigma^2$ and $Y$ is an exponential random variable with mean $\lambda$ then what will be the CDF of the $Z=X+Y$? Actually it will ...
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0answers
22 views

The concatenation of two independent normal vectors is multivariate normal.

I've already read this question. By the definition I have, $$\mathbf{z} = \begin{bmatrix} z_1 & z_2 & \cdots & z_n \end{bmatrix}^{T}$$ is a multivariate standard normal vector if each ...
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1answer
38 views

Find the conditional distribution of a variable?

Given that a lifetime has the cdf $F(t)=1−(1−t)^n$ for $0≤t≤1$ and some natural $n$. We wish to find the conditional distribution of the variable $T_x=T−x$, given that $T > x$ and $0 < x < ...
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1answer
36 views

Maximum likelihood estimator of $\lambda$ and verifying if the estimator is unbiased

$(X_1,...X_n)$ is a random sample extracted from an exponential law of parameter $\lambda$ Calculate the likelihood estimator $\nu$ of $\lambda$. Then, if $n=2$: establish if $\nu$ is a unbiased ...
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1answer
21 views

Poisson distribution of false alarms

I have this problem: The daily amount $X$ of burglar alarms has the Poisson distribution with parametr $\lambda>0$: $$P[X=k]=\dfrac{\lambda^k}{k!}e^{-\lambda}, k=0,1,...$$ It is known that ...
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0answers
18 views

Joint distribution of two normal marginal distributions

My question is related to the possibility of stating joint convergence in distribution from marginal weak convergence. Consider two sequences of random vectors $X_n$ and $Y_n$ defined on the ...
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1answer
12 views

Convergence rate of mean and standard deviation.

I have a random variable simulator with Normal distribution $(\mu,\sigma^2)$. I repeatedly conduction simulation. Each time, the simulation gives $N$ numbers $x_1,x_2,\ldots,x_N$. I use the $N$ ...