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

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1answer
511 views

Natural conjugate prior for bernoulli distribution

Assume we have an i.i.d. sample of $n$ observations from a Bernoulli distribution. That is, $\displaystyle{p(y_i|\theta) = \theta^{y_i}(1-\theta)^{1-y_i}} \ \ \ \ \text{for} \ \ y_i = 0, 1$ and $i = ...
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1answer
173 views

What is the probability distribution on the high, low and end values of a random walk?

Ok, suppose there's a random walk U which starts at 0 and has a variance of 1 over a time of 1. In other words, U(0)=0, and the probability density of the value of the function at U(1)=x is ...
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1answer
38 views

Days to live of a population where everyone has a 50% chance to produce 2 offsprings

Consider a population of day flies. Each one lives at most one day. On each day and for each living day fly there is a 50% chance to produce 2 offsprings (on its own). Start with one day fly. n | ...
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1answer
43 views

Finding the marginal posterior distribution of future prediction, $y_{n+1}$

Assume the following bivariate regression model: $y_i = \beta x_i + u_i$ where $u_i$ is i.i.d $N(0, \sigma^2 = 9)$ for $i = 1, 2, ..., n$. Assume a noninformative prior of the form: $p(\beta) ...
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1answer
77 views

Regarding calculating the bias of coin with uncertainty

Suppose you have a coin that you flip $n$ times and the result have $m$ heads and $n-m$ tails. How accurate can you predict the bias of the coin to be $\frac m n$? I know that ...
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1answer
287 views

Regarding probability bound of flip coins

Suppose you flip a fair coin 10,000 time how can you characterize the distribution of the occurrence of head? From the textbook, it says that $P[head>\frac{n}2 + k\sqrt{n}]$ < $e^{-{k^2/2}}$, ...
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1answer
74 views

convergence in probability of the average

I have a question: Let $(X_n)_n$ be a sequence of independent random variables satisfying that $X_n$ converge in probability to $0$, so what happened to the new sequence $(Y_n)_n$? where ...
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0answers
76 views

Joint PDF for spherical region

A sphere has a coordinate system (r, $\theta$, $\phi$) with the origin at the center of the sphere. What is the joint PDF of the r and $\phi$ coordinates, $f_{r,\phi}(r,\phi)$, for a randomly ...
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0answers
34 views

Confusion related to predictive distribution of gaussian processes

I have this confusion related to the predictive distribution of gaussian process I didn't get how the integration gave that result. What is P(u*|x*,u). Also how come the covariance of the posterior ...
2
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1answer
37 views

Valid distribution

If $N(0,\sigma^2)$ is the Gaussian distribution with mean $0$ and variance $\sigma^2$, is $pN(0,\sigma^2)$ a valid distribution ? $p$ is a constant and $0\le p\le 1$.
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1answer
409 views

(theoretical) Negative Binomial Distribution using Matlab

I was trying to solve some exercises on Matlab in order to improve my skills and I stumbled upon this question: For the (theoretical) Negative Binomial distribution with parameters r = 5, p = 0.4, ...
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0answers
522 views

Distribution of the $l_2$-norm of gaussian vector

Let $Y_k \sim N(\mu_k, \sigma_k^2)$. For $\sigma_k = \sigma$ the squared norm of $Y = (Y_1, \ldots, Y_n)$ follows the noncentral chi square distribution. What is the distribution in the general case? ...
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2answers
95 views

$P(X \ge 450)$ in Possion distribution

The number of pedestrians that cross the street in one minute has Poisson distribution $\def\Pois{\operatorname{Pois}}\Pois(8)$. Find the probability that at least $450$ pedestrians will cross the ...
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1answer
408 views

Deriving the density function of a exponential distribution

I am reading a book where the author tries to derive the density function of a exponential variable by the following form: Suppose that in a short interval of time $\Delta t$ there is a chance ...
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1answer
154 views

First Order Stochastic Dominance

I am reading up on stochastic dominance(http://en.wikipedia.org/wiki/Stochastic_dominance) and have some questions: PDF and CDF of Gamble A and B look like this. Since the CDF of A is always less ...
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0answers
72 views

1000 nuts in 500 cakes, random variable X = the number of nuts in a random cake

A baker puts $1000$ nuts the mixture for making $500$ cakes. $X$ is a random variable indicating the number of nuts in a randomly chosen cake, which is at most 5. Find the distribution of $X$ and the ...
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1answer
52 views

Adding a constant in “equal in distribution”

Why is that adding a constant in this equation, $x_B \overset {d}{=} (x_A+y)$ is equivalent to pushing some of the probability mass to the left if $ y \le 0$. Should it be pushing it down ...
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1answer
55 views

Probability Density Function to Cumulative Density Function

I am reading on Stochastic Dominance (http://en.wikipedia.org/wiki/Stochastic_dominance) and few questions on PDF and CDF. The paragraph I am looking at this: Why is that $P[A\ge x] \ge P[B \ge x] ...
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2answers
66 views

Integration of Probability density function

I have a question on how this equation changes from Probability Density Function to Cumulative Density Function. The equation $\int U(W)[f(W)-g(W)]\mathrm dW > 0$ changes into $\int U(W) ...
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1answer
448 views

Show multivariate Beta integrates to 1

I am trying to either find or construct a concrete example of a multivariate Beta distribution (Dirichlet) that integrates to $1$. From the definition of the Beta distribution, we have $$ \int ...
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1answer
44 views

Does the probability distribution associated with this pdf have a name?

The pdf is $$f(x)=\frac{ab^a}{(x+b)^{a+1}}$$ for $x\geq0$ and some parameters $a,b$. I've come across it a couple of times in study material for the actuarial exams.
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604 views

Maximum and minimum of an integral under integral constraints.

Find the maximum and minimum of the following integral in terms of $f(x),a,C$: \begin{align}I=\int_{0}^{a} \frac{x}{f(x)}p(x)dx \end{align} s.t.: 1) $\int_{0}^{a} p(x)dx=1$ 2) $\int_{0}^{a} ...
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3answers
446 views

Independence and Joint PDF

I know that as a general property, if $f_{X,Y}(x,y)=g(x)h(y)$, then we say X and Y are independent random variables. However, I am having trouble to accept this statement. Take the case: ...
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1answer
32 views

Combination of two Central Limit Theorems

Suppose $\{X_n\}$ and $\{Y_n\}$ are independent sequences of postive valued random variables. Furthermore, assume there exist constants $\mu_1$, $\mu_2$, $\sigma_1$ and $\sigma_2$, such that ...
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1answer
1k views

Let X be a Random Variable with probability distribution?

Let X be a Random Variable with probability distribution X -1, 0, 1, 2, 3 F(x) .125, .5, .2, .05, 0.125, a) Find E(x) and V(X) b) find the ...
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1answer
98 views

Help writing Dirichlet (multidimensional Beta) PDF correctly

I am not getting a PDF when I attempt to express the Dirichlet distribution over the random variable vector $\mathbf{\theta}=(\theta_1, ..., \theta_{27})$. Suppose a total of $2000$ observations on ...
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1answer
78 views

Closed-form expression (or good upper bound) for $\mathbb{E}\left[|X-\mathbb{E}X|^{\alpha}\right]$, where $X$ is binomial?

I am struggling to get either a closed-form expression, or as tight an upperbound as possible, for the quantity $$ M_\alpha(X)\stackrel{\rm{}def}{=} \mathbb{E}\left[|X-\mathbb{E}X|^{\alpha}\right] $$ ...
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2answers
1k views

Probability density function for radius within part of a sphere

I would like to find the probability density function for radius within a given section of a sphere. For example, suppose I specify $\pi / 4 < \theta < \pi / 3$ and $\pi /7 < \phi < \pi ...
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2answers
346 views

Probability Distribution for Product of two Random Numbers?

If I multiply two random numbers from 1 to 10 together I will get an outcome from 1 to 100. Over a large enough sample size the outcomes will tend to cluster closer to 1 than to 100 (only 21 of the ...
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70 views

Are these two distributions identical?

Let $X=\sum_{i=1}^r \lambda_{i}\|\sum_{n=1}^Ng_n\alpha_{n,i}\|^2$ and $Y=\sum_{i=1}^r \lambda_{i}\sum_{n=1}^N\|g_n\|^2\|\beta_{i}\|^2$, where $\alpha_{n,i}$'s, $\beta_i$'s and $g_n$'s are all i.i.d ...
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1answer
61 views

How Do I Get This Joint Density Function?

Given $X \sim u(0,1)$, we define $Y=1-X$, then we have that $f_{X}(x)=I_{[0,1]}(x)$ and $F_{X}(x)=xI_{[0,1]}(x) + I_{(1, \infty)}(x)$. I know, if $0\le y \le 1$ $$F_{Y}(y)=P[Y \le y]=P[1-X \le ...
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2answers
404 views

Shift interval of log-normally distributed latin hypercube samples

first of all I'm not sure if this part of StackExchange is the right one because my question is mainly on a way to implement something in MATLAB. Ok, now let me try to pack my whole question in one ...
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2answers
130 views

Urn problem: replacing white balls with black once selected

I'm trying to find the probability of an outcome where, using the traditional example, white balls are replaced by black balls once selected. Initially I have $n$ white balls and $\mu$ samples. I ...
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0answers
782 views

Why does the likelihood function for the Poisson distribution integrate to $1$?

The probability mass function for the Poisson distribution with parameter $\theta$ is $$ \mathbb P(N=n;\theta)=\frac{e^{-\theta}\theta^n}{n!} $$ Since this is a probability mass function, we have: ...
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81 views

How to generate Poissonian Gaussian distribution from sum of random variables

I want to generate a Poissonian-Gaussian distribution with $Z \sim \mathcal{N}(\mu, \Sigma) + \mathcal{P}(\lambda)$, from sum of independent random variable: $Z = \sum_i{X_i}$. Is there any easy ...
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134 views

sum of poisson and gaussian random variable

If $X \sim \mathcal{N}(0,1)$ and $Y \sim \mathcal{P}(\lambda)$ is there any name for the distribution of $Z = Y + X $ ?
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1answer
879 views

Summing Laplace random variables

If I have $N$ independent random variables, all identically distributed according to a Laplace distribution with mean $0$ and variance $\sigma^2$ (or a scale parameter of $\sigma/\sqrt{2}$), will ...
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1answer
161 views

Probabilities of uncountable intersection of events

In order to determine a probability for some event $A\in\Omega$, I ended up with $$ \mathbb{P}\left(X_t>f(t),\quad \forall [0,T]\right)≤ \mathbb{P}(A)≤\mathbb{P}\left(X_t≥f(t),\quad \forall ...
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1answer
56 views

Is there a multivariate t distribution with four parameters?

In the appendix A.2 of this paper (page 27), a multivariate t distribution with 4 parameters is mentioned. This definition is a little different from what I have seen. for example see Wikipedia . ...
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1answer
410 views

Probability generating function for logarithmic series distribution, support $k\geq1$

I'm trying to derive the probability generating function (pgf) for the logarithmic series distribution, and not getting the expected form $\frac{\log{(1-qs)}}{\log{(1-q)}}$. It seems that pgfs are ...
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1answer
60 views

Counter example required

I am searching a counter example, if there exists one for this problem. Given $4$ densities $f_0$, $g_0$, $g_1$, $f_1$, with cumulative distributions $F_0(y)>G_0(y)>G_1(y)>F_1(y)$ and the ...
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0answers
128 views

Conditional Expectation and Conditional Variance of three normal variables

$u$ is a normally distributed variable with mean $m$ and variance ${s_1}^2$, $y_1=u+e_1$, $y_2=u+e_1+e_2$ where $e_1$ and $e_2$ are independently and normally distributed with zero means and variances ...
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1answer
139 views

Calculating demand distribution of multi-step process

It's been years since I've had to work with probabilities and I need to model a business problem, my situation is this: users are signing up for a service and I need to estimate future service ...
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71 views

Distribution of scalar product of correlated unitary vectors

I have two $N$-dimensional complex normal Gaussian vectors $x,y$, i.e., each element in $x$ and $y$ is complex, with mean 0 and complex variance 1. Moreover, they have a certain complex correlation ...
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2answers
131 views

If $X$ and $Y$ are identically distributed, are $X^2$ and $Y^2$ identically distributed?

If $X$ is distributed like $Y$, can we conclude that $X^2$ is distributed like $Y^2$? I don't know how to prove it or to give a counterexample. Thanks
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1answer
80 views

Synonym for a Symmetric Bernoulli Trial?

I am wondering if there are any technical or informal, simplified synonyms for a symmetric Bernoulli Trial, $B(1,p)$, where $p=q=(1-p)=0.5$? The closest one I can think of is a "fair coin", or ...
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1answer
94 views

consistent estimator

Suppose we have $X_1, X_2,X_3, ..., X_n$ independent with unknown $\theta_1, \theta_2$ with $X_i$ normally distributed with mean $\theta_1 + \theta_2 x_i$ where $x_i$ are known and variance 1. Can ...
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2answers
307 views

Probability Distribution. Case study with a bacterial population

Let's imagine, we start with one single bacterium. At each time step (generation), each bacterium has $x$ offspring and it dies (semelparous species). $x$ is a value drawn from a normal distribution ...
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2answers
308 views

Distance between the product of marginal distributions and the joint distribution

Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose: \begin{align} P1(A,B,C) &= P(A) P(B) P(C) \\ P2(A,B,C) &= P(A,B) P(C) \\ P3(A,B,C) &= ...
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51 views

Property of $F_{m,n}$ distribution

Suppose $X \sim F_{n,m}$ and I want to show that $X^{-1} \sim F_{m,n}$, where $F_{m,n}$ is the $F$ distribution with $m$ and $n$ degrees of freedom. I had two methods of showing this which I tried; ...