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

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1answer
12 views

Gamma-Poisson Conjugacy

Busses arrive at a certain bus stop according to a poisson process with rate $\lambda$ buses per hour, where $\lambda$ is unknown. The uncertainty about $\lambda$ is quantified using the prior ...
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1answer
43 views

Binomial distribution, when variable isn't x

I've been using the formula $$p(x,N)=\frac{N!}{(\frac{N+x}{2})!(\frac{N-x}{2})!} p^{1/2(N+x)} q^{1/2(N-x)}$$ to determine the probability for a dog who walks in a straight line and can either move ...
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1answer
18 views

Let $X,Y \sim \operatorname{Expo}(\lambda)$ i.i.d, and $T = X + Y$, $W = X/Y$. Find joint and marginal PDF of $T$ and $W$

Let $X$ and $Y$ be i.i.d. $\operatorname{Expo}(\lambda)$, and transform them to $T = X + Y$, $W = X/Y$ . (a) Find the joint PDF of $T$ and $W$ . Are they independent? (b) Find the ...
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1answer
26 views

Distribution of the power of an exponential random variable

Three students are working independently on their probability homework. They start at the same time. The times that they take to finish it are i.i.d. random variables $T_1$, $T_2$ , $T_3$ with ...
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1answer
36 views

The value of $P\bigl(|X-Y|>6\bigr)$

Let $(X,Y)$ be two dimensional random variable such that $E(X)=E(Y)=3$ & $var(X)=var(Y)=1$ & $cov(X,Y)=\dfrac{1}{2}$. Then , $P(|X-Y|>6)$ is : (a) less than $\dfrac{1}{6}$ (b) equal to ...
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1answer
449 views

How to choose between poisson and binomial distributions

I don't get this thing... I know that binomial distribution is used to know the probability of a X v.a. that sounds like this: X = "the probability of having 4 blue balls doing 10 extraction from a ...
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1answer
36 views

Sum of a sum of regular normal distribution

Can someone please confirm if the solution below is correct? If not, please give me some hints about what is wrong. Let $X$ and $Y$ be two real valued stochastic variables who's joint distribution is ...
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1answer
37 views

Probability of $P\bigl(X+Y<\frac{1}{2}\bigr)$.

Let $X$ & $Y$ be two continuous random variables with the joint probability density $$f(x,y)=2 ,0<x+y<1,x>0,y>0$$ $$f(x,y)=0,elsewhere$$ Find the value of ...
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1answer
24 views

standart Cauchy distribution

Suppose that X has a standard uniform distribution. Show that Y=tan(2πX) has a standart Cauchy distribution.
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33 views

Probability distribution of an area

I've got the following problem: Given an equilateral triangle of height $1$, let's draw two lines from one of its vertices with random directions (uniform). Let's consider the half lines of those ...
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0answers
19 views

Conditional distribution on the unit circle and a square

Let (X,Y) be uniformly distributed over $B=\{(x,y) \in \mathbb{R}^2: x^2+y^2 \leq 1 \}$ resp. $Q=[-1,1]^2$. Now I want to calculate the conditional distributions and of Y given X=x. And then the ...
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1answer
109 views

Expected value of function of negative binomial

With $X$ representing the total number of trials, and m the fixed number of successes. The pdf is then $f(x|p)=$${x-1}\choose{m-1}$$p^m (1-p)^{x-m} \ \ \ \ x \ge m$ As a step in something else I'm ...
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0answers
21 views

Posterior distribution of two independent random variables with distinct Beta distributions

$\def\Beta{\operatorname{Beta}}$ If $X,\ Y\sim \Beta(\alpha, \beta)$ and $x$ is a vector, then $ P(X>Y\mid x) = \iint_{X>Y}P(X,Y\mid x) \,dX\,dXY $ I need to compute $P(X>Y\mid x)$ when ...
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0answers
35 views

Probability distribution based on random events with increasing weights

Consider a weighting function, f(t), that is monotonically increasing with the time, t. [This may not be strictly necessary, as pointed out in the comments, but all of the cases I am actually ...
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0answers
12 views

Minimising the expected value

Given a Gaussian distribution, I sample a value, say $\alpha$ from the the normal distribution and then i use it as an input to a complex neural network-like function. Based on each $\alpha$, I get an ...
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0answers
19 views

Nakagami Random variable with shape parameter $m=\infty$.

A Nakagami fading distribution $X$ with parameter $m$ is given by the following $$ X\sim f(x;\,m,1) = \frac{2m^m}{\Gamma(m)}x^{2m-1}\exp\left(-mx^2\right)$$ Then the function $S:=|X|^2$ is Gamma ...
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1answer
40 views

Weak Convergence If and Only If (Pointwise) Convergence of Characteristic Function

This is actually a theorem from lecture notes, with the corresponding proof. Unfortunately, it doesn't prove the last bit, or mention it at all (!), and I have a question about the penultimate bit. ...
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2answers
99 views

The distribution of the minimum of two independent geometric random variables

Let $X$ and $Y$ be independent geometric random variables. What is the distribution of $Z=\min(X,Y)$? The probability mass functions are $\operatorname{Pr}(X=k)=(1-p)^{k-1}p$ and ...
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1answer
317 views

Unknown number of colours Bernoulli Urn

Okay, so, in the traditional Bernoulli Urn problem, we have an urn with a number N, possibly infinite, of coloured balls, and there are k possible colours. That one I grok. However, what if I don't ...
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0answers
35 views

Poisson-Binomial distribution approximated by binomial distribution

I am looking for strategies how to approximate poisson-binomial distribution (PB) via the binomial (B) distribution. I have seen a few papers [Ehm91,Roos01,LeCam59] on them. The papers uses total ...
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1answer
31 views

Question about sum of chi-squared distribution

I want to prove that the sum of two independent chi-squared random variables is a chi-squared random variable. I am supposed to only use the fact that if $Q$ has a chi-squared distribution with ...
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0answers
18 views

Iso-density locus of Gaussian mixture distribution

I would like to known what is the equation of the iso-density surface of a Gaussian mixture distribution. Is such an iso-density surface a union of ellispoids? Let's say that this Gaussian mixture ...
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1answer
27 views

To find the probability of the event associated with the binomial distribution

Let $(X_i)_{i \in \mathbb{N}}$ independent random variables which have a Bernoulli distribution: $$X_i = \{0, 1\},\;\mathbb{P}(X_i = 1) = p,\;\mathbb{P}(X_i = 0) = q, \; p + q = 1$$ And define: ...
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0answers
34 views

How to answer this question? Mgf of a gamma

I am revising for an exam and this question has appeared and I don't know how to tackle it. In previous parts we are asked to work out the mgf and mle of a gamma which I have done and they are ...
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1answer
41 views

Multivariate normal distribution. To find the distribution of the random vector.

Let $(\eta_1, \eta_2, \eta_3) \sim N_3 (\mu, A)$, where $\mu = (0,0,0)$ and $A = \begin{bmatrix}2 & 1 & 0\\1 & 1 & 0\\0 & 0 & 1\end{bmatrix}$. Need to find the distribution of ...
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1answer
298 views

Let (X,Y) have a Dirichlet Distribution with paramters $(\alpha_1, \alpha_2, \alpha_3)$ Establish that X~Beta$(\alpha_1, \alpha_2 + \alpha_3)$

If the joint pdf of (X,Y) is $f(x,y)=\frac{\Gamma(\alpha_1 + \alpha_2 + \alpha_3)}{\Gamma(\alpha_1) \Gamma(\alpha_2) \Gamma(\alpha_3)} x^{\alpha_1 - 1} y^{\alpha_2 - 1} (1-x-y)^{\alpha_3 -1}$ ...
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0answers
31 views

Show Almost Certain Convergence of a Sequence of Normal Random Variables

Let $(X_n)_{n=1}^\infty$ be independent, $N(0,1)$-distributed random variables. Prove that $$ \limsup_{n \to\infty}{X_n \over \sqrt{2 \log(n)}} = 1 \ \text{almost surely}.$$ I am aware of the ...
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0answers
39 views

Convolution of a pdf $f$ with a Gaussian $g$: distance between $g\ast f$ and $g$?

I have been looking for references on the following matter: let $f$ be the pdf of any real-value random variable ($f$ is not necessarily continuous wrt Lebesgue measure), and $g=g_{\mu,\sigma}$ be a ...
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1answer
268 views

Find the MOM estimate and the MLE of the Pareto distribution.

The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: Assume that $X_0$ > 0 is given and that $X_1, X_2, ..., X_n$ is an i.i.d. sample. ...
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1answer
16 views

distance-measure method to measure the distance between two matrixes(probability distribution)

I should find a suitable distance-measure method to measure the distance between two matrixes. The elements of such matrix is 0 to 1, and the sum of the all element is 1, so I think I could treat it ...
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0answers
42 views

Is time a continuous random variable?

I am trying to derive the equation for the mean value of a function, the standard formula is given by: $$\frac{1}{T} \int^T_0{g(t)dt}$$ To do this I have used the law of the unconscious statistian ...
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1answer
42 views

What is the possible random variable?

Here is a probability problem from Ross. Suppose that a bus starts from point A and reaches point B, covering a total of 100 miles between A and B. Suppose also that it suffers a breakdown on its ...
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1answer
59 views

How to show that a closed ball doesn't allow stochastical ordering

Given a closed ball $$\cal{F}=\{g:D(g,f)\leq\epsilon\}$$ where $f$ and $g$ are some density functions and $D$ some distance say relative entropy: $$D(g,f)=\int ...
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1answer
33 views

How to find the probability of such events?

I solve the following problem. Let $\xi_1, \xi_2, \xi_3, \xi_4$ - independent random variables having a normal distribution with a parameter $(0, 1)$. This means that values $\xi_i$ have a p.d.f. ...
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1answer
30 views

Expected value of power of exponential order statistics

Finding the expected value of the $\gamma$:th power of the $k$:th standard exponential order statistic in a sample of $n$ implies evaluating the integral: ...
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1answer
19 views

Finding a joint distribution

Let $(T_n)_{n \geq 0} $ be a sequence of random variables such that $T_0 =0$ and $(T_n -T_{n-1})_{n \geq 1}$ are independent exponential random variables with parameter $\theta >0$. Can someone ...
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0answers
72 views

How do I solve the following integration?

The integral gives probability distribution of $M$, and $M$ is the absolute value of the sum of two random variates both following $P(\mu)$ distribution.
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2answers
52 views

Let $U, V \sim U(0,1)$ be independent. What is $P(U \leq V)$?

Let $U, V \sim U(0,1)$ be independent. What is $P(U \leq V)$? My attempt: We are looking for $P(U-V \leq 0)$. For a given $t \in [0,1]$ this is equivalent to $$P(U \leq t, V \gt t)= P(U \leq t)P(V ...
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1answer
590 views

Correlation between Beta distributions

I have a Computer Science background and not very knowledgeable in Probability and Statistics. So excuse me if my question,notation, or language is flawed. Anyways, the problems is that we have two ...
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1answer
562 views

Great Monty Hall application in real life?

Suppose you are doing a multiple choice question with 4 different answers you have no ideas about. You mentally choose one (say A), and as you are about to write that down... you suddenly remember 2 ...
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3answers
47 views

Graphic of the probability distribution function : How does it works?

Here is the graphic of the probability distribution function for a random variable $X$. How can I find the distribution of $Y=-X$? By definition the distribution function for a random variable is ...
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0answers
42 views

Minimum distance estimation approach in inference

I am having a problem with a very basic concept in the minimum distance estimation approach in statistical inference. I've read a paper which uses, for a parametric model family of discrete ...
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0answers
27 views

how can I find outliers in a 2 vector data set

I have a two data set $(X,Y)$ where $X$ represents the angles and $Y$ represent the signals. $X$ is always correct because I increment it by coding $x=x+1$. However, $Y$ could be sometimes wrong ...
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1answer
21 views

Carrying out the passage to the limit under an integral sign

For a sequence of distribution functions $(F_n)$ and their characteristic functions $(\varphi_n)$ I got $$ ...
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0answers
33 views

Distribution and density functions, and upper bound limit

I am working on some functions, such as distributions and density functions, and upper bound. CASE I My first function: $\forall x\in\left[0;A\right]$, $f(x)=ax(A-x)$. The question is: for which ...
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1answer
362 views

joint probability distribution of one discrete, one continuous random variable

This is a problem on the joint distribution of a discrete and a continuous random variable. Kitty Oil Co. has decided to drill for oil in 10 different locations; the cost of drilling at each ...
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1answer
25 views

Find the marginal distribution of a bernouilli and continuous joint distribution

I need to find the following. Is the method right? Find $f_X(x), f_Y(y)$ if the joint distribution in $(x,y)$ is given: $$\frac{p^x(1-p)^{1-x}}{\sqrt{2\pi}\sigma} \exp \left( ...
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1answer
40 views

Probability distribution function of $Z=\frac{Y}{X+1}$

I have two random variable $X,Y$ in the same space $(\Omega,\mathcal{F},\mathcal{P)}$. $X$ takes its values on $\Bbb{N}$, I denote $P(X=k):=p_k$ and $Y$ takes its values on $\Bbb{R}$? and ...
2
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0answers
35 views

Distributions of infinite sums of random variables

Let $(Z_i)_{i \in \mathbb{N}}$ be the arrival times of a Poisson process of intensity $1$ on the interval $[0,\infty)$, i.e. $Z_i \sim \text{Gamma} (i,1)$. We define $X = \sum _{i \in \mathbb{N}} ...
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0answers
24 views

Is this an equivalent definition of probabilistic independence of random variables?

Let $X$ and $Y$ be random variables, $P(X=x) \neq 0$, $P(Y=y) \neq 0$ for any $x,y \in I\!R$, and suppose that \begin{equation} P(X=x ~|~ Y=y) ~~~=~~~ P(X=x ~|~ Y=z) \end{equation} holds for all ...