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

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Given the sum of four Exp(1) distributed random variables, what is the conditional density of sum two of them?

Let T := X+Y+Z+K be indepedent and Exp($1$)- distributed random variables. What is the density of (X+Y) given {T = $1$} ? For M:= X+Y and N := Z+K given {M + N = $1$} The joint density is $ ...
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1answer
13 views

Various modes of convergence of random variables

Let $\lbrace X_n \rbrace_n$ be a sequence of independent random variables such that $$P(\{X_n = \pm 1 \}) = \frac{1}{n}$$ $$P(\{X_n = 0 \}) = 1 - \frac{2}{n}$$ Is the sequence convergent: $1$) almost ...
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0answers
8 views

Mean Preserving PDF Spreading

I have a univariate discrete random variable and a histogram representing its PDF (which is asymmetrical). Is there a known way to increase/decrease the variance of the distribution (i.e. scaling it ...
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1answer
22 views

Given the distribution of $X$ and $Y=-2\theta \ln X$. How is $Y$ distributed?

The pdf of $X$ is $f(x) = \theta x^{\theta-1},\enspace 0<x<1, \enspace 0<\theta<\infty.$ Let $Y=-2\theta \ln X.$ How is $Y$ distributed? My work: $$ \begin{align*} F(Y) = P(Y \leq y) ...
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2answers
94 views

Who first found the value of $\int_{-\infty}^{+\infty}e^{-x^2}dx$?

A fairly pretty technique of showing that $$\int_{-\infty}^{+\infty}e^{-x^2}dx = \sqrt{\pi}$$ is to square the integral, writing that square as the product of two integrals with integration variables ...
1
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1answer
15 views

Expression defined by exponential random variables, probability of being nonnegative

Consider $n \geq 2$. Let $E_1,...,E_n,F_1,...,F_n$ be independent exponentially distributed random variables with rate $1$. Define $T_E = \displaystyle \sum_{i=1}^{n}{E_i}$, and $T_F = \displaystyle ...
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2answers
27 views

Calculating $E[X|Y]$ for continuous $X$ and discrete $Y$

I'm struggling with the following exercise, which I have the solution to but don't understand. I would appreciate any help. The exercise Let $X$ a random variable with $f_X(x) = 2x$ if $x \in [0,1]$ ...
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39 views

A student only wrote down …

In the statistics lecture $6$ discrete and $5$ continuous distributions were discussed. For each distribution one can ask for $\mathbb{P}(X = a), \mathbb{P}(X \leq a), \mathbb{p}(X \geq a), ...
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0answers
14 views

Find the number of items in $10000$ sets of 10 throws each in which you would expect no even numbers.

Given to us is that we have an irregular six-faced die and the expectation that in $10$ throws, $5$ even numbers show up is twice the expectation that $4$ even numbers show up. The question( as in the ...
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0answers
20 views

Central limit theorem with Lyapunov condition

$Z_1, Z_2,...$ are iid uniformly distributed on $[-1;1]$, $\lim_{n \to \infty} a_n=0$ and $\lim_{n \to \infty} na_n=\infty$ also $a_n>0$ $\forall n$, $X_{n,j}= \frac{1}{a_n}I(|Z_j| \le a_n)$ ...
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0answers
17 views

Computation of two-sided probability density functions from their cumulants using Laplace transform

The computation of one-sided probability density functions (PDFs) from their cumulants using Laplace transform is proposed by following paper: M.N. Berberan-Santos, Journal of Mathematical Chemistry, ...
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1answer
14 views

Is it possible to evaluate a normalizing constant for a characteristic function

Let $X$ be a random variable with density $f$ and characteristic function $\varphi$. Say we know $\varphi$ up to a constant $c$. Is it possible to evaluate this constant using $\int f(x)dx=1$ (or by ...
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0answers
21 views

Conditional probabilities given the evidence(Bayesian network)

Let's say we have a Bayesian network: How can I compute P(A | F, E) ? I have all the probabilities for each node. Thanks!
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2answers
44 views

How to compute the sum of geometric distribution [on hold]

How to compute the sum of random variables of geometric distribution $X_{i}(i=0,1,2..n)$ is the independent random variables of geometric distribution, that is, $P(X_{i}=x)=p(1-p)^{x}$, then how to ...
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1answer
13 views

Simplifying this summation

I've been doing this question and I'm stuck! Each customer who enters Larry’s clothing store will, independently of every other customer, purchase a suit with probability p. Assume that N, the ...
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0answers
21 views

How to “reduce” a probability distribution satisfying certain conditions

I will try and explain the question I have in term of an example. I am given some probability distribution $f$, in this case of 2 variables x and p, $f(x,p)$. For example, I can pick the ...
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2answers
11 views

metrics for density-sampling similarity, beyond likelihood

I am looking for a metric that would evaluate the distance between a sample $S$ and a density function $D$ Building a sample from a known distribution can be done using a monte-carlo sampling, ...
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2answers
26 views

Convolution of 2 uniform random variables

I really do not know how to do this. Let $X$ have a uniform distribution on $(0,2)$ and let $Y$ be independent of $X$ with a uniform distribution over $(0,3)$. Determine the cumulative distribution ...
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0answers
14 views

Restricted Boltzmann Machine Derivation

From book chapter, RBM probability is shown as $$ P(x,h;W) = \frac{1}{Z(W)} \exp \bigg[ \frac{1}{2} y^T W y \bigg] $$ wnere $y \equiv (x,h)$ The book mentions after maximum log likelihood, he ...
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9 views

normalization of probability density on a surface

Let $p(x)$ ($x\in \mathbb{R}^d$) be a probability density function defined on $\mathbb{R}^d$. Assume we have a closed surface $S\subset \mathbb{R}^d$, on which points follow this function $g(x) = 0$. ...
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0answers
10 views

Can particular outcomes from continuous random variables be said to be probable/improbable?

Suppose a continuous variable $x$ is randomly distributed. For concreteness, let us say that it is Gaussian distributed, $x\sim N(0,1)$, such that $p(x)=\textrm{Gauss}(x; 0, 1)$. From $x$, I can ...
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0answers
10 views

The expected number of mutations in a sequence of elements, each with random delays

In a sequence, the number of the permutations, is the (minimum) number of the pair of elements needed to switch to make them sorted. For example in the following: ...
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16 views

Convergence in distribution plus convergence of moments.

Suppose that the sequence of r.v $\{X_n\}_{n\geq 1}$ has all the moments, and $X_n\stackrel{D}{\longrightarrow} X\sim N(0,\sigma)$. Assume that $E\{(X_n)^K\} \stackrel{n} {\longrightarrow} E(X^K)$, ...
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28 views

Estimate probability density function of being in a certain time interval

​You arrive at a bus stop in an unfamiliar part of town. Assume that buses arrive at the stop with an unknown (to you) distribution and wait in the bus stop for a few ​minutes. The wait time ...
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0answers
16 views

Comparing infinite dimensional distributions

Given two infinite sequences of rvs $(X_{1},X_{2},...)$ and $(Y_{1},Y_{2},...)$, how can we show $(X_{1},X_{2},...)\stackrel{d}{=}(Y_{1},Y_{2},...)$? The way I heard is by comparing all their finite ...
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0answers
17 views

ML estimation for Weibull

What are the maximum likelihood estimators of $\eta$ and $\beta$ ($\eta>0$ and $\beta>0$) for an i.i.d. sample of size $n$ from the following density: $f(y_i)=\frac{\beta x_i^{\beta-1} }{\eta ^ ...
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1answer
27 views

Calculate probability of joint PDF

I'm given the following joint PDF and asked to calculate $P(X+Y>1)$ $f_X$$_Y$$(x,y)=2/5$ for $0<y<1$ & $0<x<5y$ and $f_X$$_Y$$(x,y)=$ $0$ else I know I have to take the ...
3
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0answers
54 views

Using Jensen's inequality to prove the Cauchy distribution has no mean

I can see that there is no mean because $\int x / \pi(1+x^{2})$ does not converge from -inf to inf. But my prof hinted at using Jensen's inequality for the proof. $$f(E(X)) \le E(f(X))$$ How can I ...
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1answer
37 views

How to understand $\partial^2F(x, y) \over \partial x \partial y $ = $f(x, y)$

$\partial^2F(x, y) \over \partial x \partial y $ = $f(x, y)$ --------------- (1) This formula comes to me but I have no idea what it means. Especially what the $\partial^2$ and $\partial x$ ...
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1answer
29 views

Birthday paradox derivation: different approach

I usually use randomization in algorithms so I am familiar with basics of probability but nothing much advanced. I have gone through the derivation for Birthday Paradox (Cormen et al) and decided to ...
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0answers
7 views

Find continuos distribution for a given distribution (maybe discontinuos )

Let distribution $F$ is regular $F(F_*-) = 1$ and $\lim_{x\to F_*-}\frac{1-F(x)}{1-F(x-)}=1$, $F_*=\sup\{x:F(x)<1\}$ Find continuos distribution $G$ such that $\sup_{x \in R}{|F(x)^n-G(x)^n|}\to ...
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1answer
16 views

Building a compound probability distribution

I want to build a probability distribution for a "shock" variable. I want to show that there are p% chances of no shock, and (100-p)% chances of shock- in which case, shock is distributed according to ...
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0answers
18 views

How to find mean and variance for probability problem with warranty?

I am in a probability theory class and I'm stumped on a problem: A warranty is written on a product worth \$10,000 so that the buyer is given \$8000 if it fails in the first year, \$6000 if it fails ...
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0answers
13 views

Kullback-Leiber divergence for two simple probability vectors

For any probability vectors $ p= (p_1,...,p_K) $ and $ q=(q_1,...,q_K) $ representing monotonically increasing functions $ x-1 $ and $ ln(x) $ respectively what is the KL divergence? $$ KL(p||q) = ...
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1answer
29 views

Probability density function of $max(X,Y)$

Assume we have random variable $W = \max({X,Y})$, I would like to find the pdf of $W$. This is what I have done. $$ F_W(w)= \mathbb{P}[ W\leq w]=\mathbb{P}[ \max({X,Y})\leq w]=\mathbb{P}[ X\leq ...
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0answers
32 views

How to approach solving multi-variable continuous probability distrobution problem

You are taking the subway in an unfamiliar city. You are told to take the Blue Line train to central station and then transfer to the Green Line train, which is just on the other side of the platform. ...
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1answer
10 views

Convergence in distribution of a normalized Poisson distributed random variables

Show using the central limit theorem that $\frac{X_n-n}{n^{1/2}}\rightarrow Z$ where $Z$ is standard normally distributed and $X_n$ is $Poisson(n)$ distributed.
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1answer
19 views

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$ i) Find the regions where the joint pdf of $(Z,W)$ is positive. ii) Find the ...
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1answer
17 views

An example of covergence to an exponential distribution, the role of continuity

I got a probability problem I can solve, but my solution does not use an assumption which is given in the formulation of the problem. I am afraid that this is might be a sign that my solution is ...
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2answers
31 views

Why would a uniform prior distribution give a different result than a purely frequentist approach?

I would expect a uniform prior to be a good example of an uninformed prior and get the same result as the frequentist approach. However, this is not the case. As an example, let's look the classical ...
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1answer
20 views

Showing independence of two random variables

The problem is here The trouble im having is showing how $\bar{x}-\bar{y}$ is independent of $S_{pool}$. I know the covariance of ( $\bar{x}-\bar{y}$,$X_i-\bar{x}$)=0 and similarly for the other ...
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15 views

distribution of the maximum of independent poisson random variables.

Let $X_i$ $i=1,\dots,n$ be independent poisson random variables with $X_i \sim \text{Poisson}(\lambda_i)$ then we define $X = \max_i X_i$ how does $X$ distribute? Is easy to see that ...
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2answers
33 views

linear combination of random variables

Let $X$ and $Y$ be $iid$ uniformly distributed random variables over the interval $[0,1]$. We know by convolution that the distribution of $Z=X+Y$ is given by: $$f(z) = \left \{ \begin{array}{ccc} z ...
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15 views

The distribution of the sum of a uniform random variable and a binomial random variable

I'm asked to find the distribution of $U=X+Z$, where $X\widetilde~R(0,1)$ - That is, $X$ has a uniform distribution for $x\in]0;1[$ $Z\widetilde~bin(1,1/2)$ - That is, $Z$ has a binomial ...
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0answers
10 views

Wasserstein distance and maximization covariance

My question deals with the second order wasserstein distance $W_2$ on the set of measures, which is defined by: $W_2(\nu_1,\nu_2)^2= inf_{\Pi(X,Y)} E_{\Pi} (X-Y)^2$ where $\Pi$ is chosen such that ...
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83 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
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2answers
24 views

What's an intuitive description of the meaning of standard deviation in a discrete uniform distribution?

Just starting out with distributions, so I'm looking for an every day explanation to help me understand. I've read that for a discrete uniform distribution, the standard deviation is a measure of the ...
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4 views

Divisibility property of Laplace

What does it mean that the laplace distribution has the divisibility property?Is there any distribution that this does not hold?
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10 views

Aggregation Urn Distribution

I am trying to identify this distribution in terms of the number of balls, $n$, urns, $m$, and iterations, $i$. Before the first iteration each ball is independent. The first iteration consists of ...
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1answer
22 views

Conditional Distributions

Choose a random integer $X$ from the interval $[0, 4]$. Then choose a random integer $Y$ from the interval $[0, x]$, where $x$ is the observed value of $X$. Make assumptions about the marginal pmf ...