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

learn more… | top users | synonyms

0
votes
0answers
5 views

Fitting power law to existing integral

I have empirical data - people from cities - a certain number of people for a certain number of cities. I know the exact number of cities, as well as the exact number of total people - e.g. the ...
0
votes
0answers
14 views

Lognormal approximation of the sum of successive values of a lognormal process

I would like to use a lognormal process to approximate the successive values of another lognromal process. Let $X_t$ be a lognormal process. I would like to approximate $$ Y_t := \sum_{t=0}^T X_t $$ ...
1
vote
0answers
28 views

A model to describe probability to win at certain skill ranges?

Let's say we have a list of all the chess players in the world, and we want to predict the likelihood of success if any player goes up against any other player. (Hypothetical example) I'm assuming ...
0
votes
2answers
13 views

Can a geometric random variable have a finite sample space? [closed]

Can it be finite? I think it has to have an infinite sample space (according to my lecture notes)
0
votes
1answer
74 views

Probabilities of errors in three independent transmissions

i have been working through some old exam papers and have gotten stuck on this last one. can anyone help? When a piece of information (a bit) is transmitted over a communications channel, it may be ...
0
votes
2answers
40 views

95% Confidence Interval Problem for a random sample

The sample mean of a random sample of $25$ observations is $9.6$ and the sample variance is $22.4$. Derive a $95$ confidence interval for the population mean. I calculated the following: Confidence ...
1
vote
0answers
17 views

Higher order terms in Taylor expansion tend to infinity faster.

Suppose $g$ is a smooth bounded and symmetric probability density function (pdf). Let $\{(X_1,Y_1), ..., (X_N,Y_N)\}$ be a random sample from the joint pdf $t(x,y)$. Further assume $a\to 0$ and $Na ...
2
votes
1answer
21 views

Is $a+r \cdot b$ an uniformly random value when $a,b$ are fixed and $r$ is random value?

Imagine we have two fixed values $a,b \in \mathbb{Z}_p$ and a uniformly random value $r\leftarrow \mathbb{Z}_p$, for large prime number $p$. Question: Is $v=a+b\cdot r$ an uniformly random value in ...
1
vote
1answer
45 views

A coin probability question

Let $p$, $q$ be values in $[0,1]$ and $\alpha \in [0,1]$. Assume $\alpha$ and $q$ known, and that $p$ is unknown parameter we would like to estimate. A coin is tossed n times, resulting in the ...
0
votes
0answers
9 views

Estimating Johnson Distribution Parameters by Quantile

In this paper: http://www.researchgate.net/publication/31291960_Quantile_Estimators_of_Johnson_Curve_Parameters the four parameters for the Johnson distribution ...
0
votes
2answers
19 views

Probability Distributions (Tree Diagram)

Satish picks a card at random from an ordinary pack. If the card is ace, he stops; if not, he continues to pick cards at random, without replacement, until either an ace is picked, or four cards have ...
1
vote
2answers
25 views

Using Normal Distributions to find Proportion

The height of a randomly selected woman from a population is normal with $\mu=165cm$ and $\sigma=7cm$. The heights f the men in this population are normal with $\mu=178cm$ and $\sigma = 8cm$. I am ...
0
votes
1answer
30 views

Functions of a random variable

Assume that $Y$ ~ $Exp(Ω)$. Find the cdf and pdf of $Z$ = |$Y$ - $δ$|. In order to solve this question so far, for $Y$, I am thinking about using the pdf equation for the exponential distribution i.e. ...
0
votes
3answers
36 views

Finding the Marginal Distribution of Two Continuous Random Variables

The continuous random variables $X$ and $Y$ have the joint probability density function: $$f(x, y)= \begin{cases} \dfrac{3}{2}y^2, & \text{ where } 0\leq x \leq 2 \text{ and } 0 \leq y ...
2
votes
1answer
28 views

Poisson Approximation of Binomial

I have to prove the Poisson approximation of the Binomial distribution using generating functions and have outlined my proof here. Given, \begin{align} & \lim_{n\to \infty} np_n = \lambda \\ ...
1
vote
0answers
52 views

Need simplified formula of probability equation

I have RV $x$ which is function of independent continuous RVs $x_1$ and $x_2$. After some manipulations, I came up with an expression for the outage probability of $x$ as $$P_\text{out}(y)=\Pr(x\leq ...
0
votes
1answer
60 views

Breaking probability theory by having a different number of random variables depending on a conditioning random variable.

I suspect I'm breaking probability theory but I don't know how or why. How does one handle working with conditional probabilities where one can have a different number of random variables depending on ...
1
vote
0answers
47 views

Sufficient condition for convergence in distribution in the plane

I'm trying to show convergence in distribution for a sequence $X_n$ of random variables in the plane. Here's what I know. I have a sort of squeeze theorem for the probability of the r.v.s being in a ...
2
votes
2answers
47 views

Is $P(n) = \frac{a n }{b}$ or $\frac{(a+1) n}{b + 1}$?

I investigated Some random data and I was a bit confused. Could be Mathematical coincidence but i'm not sure. Consider the integers $1,2,3,...,a$ Randomly Pick $b$ dinstinct element out of them. ...
1
vote
1answer
19 views

If $x_1, \ldots, x_n$ have probability distribution function $F(x)$, then the maximum has probability distribution function $F(x)^n$

A random sample $x_1,x_2,.....,x_n$ is taken from a population , which has the probability distribution function $F(x)$ and the density function $f(x)$ . The values in the sample are arranged in ...
2
votes
1answer
32 views

Understanding the matrix normal distribution

A random $n \times p$ matrix $X$ is distributed according to a matrix valued normal distribution iff $\mathrm{vec}(X) \sim \mathcal{N}_{np}(\mu, V \otimes U)$, where $\mu \in \mathbb{R}^{np}$ is a ...
-3
votes
2answers
27 views

How to proof that the median of a lognormal distributions equals $\exp(\mu)$ [closed]

If $V$ is lognormal distribution, how can you prove that his median equals $\exp(\mu)$? With $\mu$ the mean of the normal distibuted $\ln(V)$
1
vote
1answer
12 views

If $E[(X \wedge b) \vee a] = E[(Y \wedge b) \vee a]$ all $a\leq b$ is $X =^d Y$?

If $X,Y \in L^1$ and $E[(X \wedge b) \vee a] = E[(Y \wedge b) \vee a]$ for all $a\leq b$ do we necessarily have $X =^d Y$? Taking $a=0$ the integrands are positive so we may use fubini to find ...
1
vote
2answers
39 views

A trivial question about prediction of arrival rate of a Poisson process from sample data

A bus arrives at a bus stop according to a Poisson process. It is given that in the last 100 hours, the bus arrived at the bus stop exactly 200 times. Predict the arrival rate for the bus at the bus ...
0
votes
0answers
6 views

How to test if an number generated from an exponentially distributed random variables is statistically different from zero at some confidence level?

For instance, let the generated number be 0.3 and the mean of the exponential distribution where this number comes from be 0.03. At a significance level of 0.10 or 0.05, is it 0.3 statistically ...
0
votes
0answers
12 views

Bernouilli-distribution, geometric-distribution conditional probability

A homework question I'm stuck with: Let X and Y be two independent discrete random variables. X has the Bernoulli-distribution with parameter $\alpha$ : $$ p_{X}(0)=1-\alpha, p_{X}(1)=\alpha $$ Y has ...
-1
votes
1answer
31 views

The CDF and PDF of the transformation of a random variable (absolute value) [closed]

Let X~Exp(λ). Calculate and find the CDF and PDF of Y = |X-μ|. So far my working on paper is here, but I get stuck on how to continue. Any suggestions would be greatly appreciated! ...
2
votes
1answer
23 views

Explanation for “jointly pdf is constant but marginal pdf is not”

Consider: $X,Y \sim \text{uniformly distributed in }(0 \leq y \leq x \leq 1)$ From short computation, we know: Jointly pdf: $f_{XY}(x,y) = 2$ Marginal pdf of $x$: $f_{X}(x) =\int_0^x ...
0
votes
0answers
16 views

Kullback leibler divergence between two language models

My question is associated with comparing two n-gram language model using KLD. Consider a 2 bi-gram language models: $P(S)= \prod_{i-1}^{l} p(w_i|w_{i-1})$ and $Q(S)= \prod_{i-1}^{l} ...
0
votes
0answers
41 views

How to find the distribution from a given form of generating function

I have the generating function defined by F(x)= $\sum P(n,s) x^n$ . And the expression for F(x) is given by $F(x)= e^{\frac{a}{b}x} (1-x)^{b/s}$. Then how can i find p(n,s) function..? Can anybody ...
0
votes
0answers
16 views

Probability Law of Stochastic Process Definition

I am reading Probability and Stochastics by Çınlar, and am confused by the following definition in it: I must be missing something because this definition does not seem correct to me. For ...
0
votes
0answers
19 views

Construct bivariate symmetric (polynomial) nonnegative functions (distributions) over the unit square with certain properties

Construct bivariate symmetric polynomials $f(x,y) = f(y,x) \ge 0$ over $[0,1]^2$, with $f(1,y) = f(x,1)=0$, such that the univariate marginal distributions are both proportional to $$(1-u^2)^4$$, ...
1
vote
1answer
28 views

Let $X$ denote the number of tosses required to get the 5th head and $Y$ the number between the 6th and 7th heads. Are $X$ and $Y$ independent?

Let $X$ denote the number of tosses required to get the 5th head and $Y$ the number between the 6th and 7th heads. Are $X$ and $Y$ independent? Y will always depend on X . NO ? i know geometric ...
0
votes
1answer
18 views

X denotes government will increase payment. x~Bin(2,2/3) . if one increment =9%. expected increment =?

If Government increases payment then they increase it by 9% . now if whether government will increase payment follows binomial distribution with parameters n=2 and p=(2/3) , then what percentage of ...
2
votes
1answer
38 views

how that $P(G)=1$ iff $\sum_n \Bbb P(A \cap E_n )=\infty$ for all events $A$ having $\Bbb P(A)>0$.

Two probability problems: 1. Let $a>0$ and let $X_n$, $n \geq 1$, be iid r.v. that are uniform on $(0,a)$ and let $Y_n = \prod_{k=1}^{n} X_k$. Determine all values of $a$ for which $\lim_{n ...
-1
votes
1answer
26 views

The motivation for considering exponential families of distributions [closed]

I saw problems of the form: "show that the distributions ... form an the exponential family". Why is this property, being an exponential family, important?
2
votes
0answers
14 views

Compute the Gibbs energy

I have a question about Gibbs distribution in Stochastic theory. In which, it defined a clique as a a subset $C$ in the whole image $\Omega$ if two different element of $C$ are neighbors. Figure 2 ...
0
votes
0answers
15 views

Probability Distribution on the Simplex with support on the faces

I am looking for a parametrized distribution on the (probability) $K$-simplex with support on its $(K-1)$-faces. I.e. say $(x_1,...x_{K+1})$ are the coordinates of the simplex with $\sum_jx_j=1$, then ...
0
votes
1answer
57 views

probability generating function for multivariate distribution

I would like to ask how can we derive PGF of any multivariate distribution? and can anyone give an example of deriving the PGF of a multivariate distribution? That will be great. Thanks advance.
0
votes
1answer
22 views

Using loss function to find Bayes estimate

I have a 2 part question, the first I believe I have figured out. The question is: Let $Y_1, Y_2, ..., Y_n$ be a random sample from a gamma pdf with parameters $r$ and $\theta$, where the prior ...
1
vote
0answers
36 views

Finding a polynomial approximation of a PDF

I would like to find a polynomial $P(x)=\sum_{d=1}^D P_dx^d$ of degree $D$, where its derivative is larger than or equal to a given pdf $f(x)$ in $[0,1-\epsilon]$, for any $\epsilon>0$. Note that ...
1
vote
1answer
60 views

Probability generating function of bivariate Poisson distribution!

Problem setup: $X_1=Y_1+Y_0,X_2=Y_2+Y_0$ where $Y_1, Y_2\text{ and }Y_0$ are independent Poisson random variables with parameters $θ_1, θ_2\text{ and }θ_0$, respectively. I know that the joint ...
2
votes
0answers
27 views

What does it means of Normalization term of Gibbs distribution?

I am studying about Gibbs distribution concept and I am confusing about the term" normalization ". According to the Hammersley–Clifford theorem, an random $x$ can equivalently be characterized by a ...
0
votes
0answers
23 views

Correspondence between AB-divergence and Kullback-Leibler divergence

I'm reading up on AB-divergence (alpha-beta-divergence) based mainly on the exposition given in Chichoki et al. (2011), "Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative ...
3
votes
1answer
28 views

For every probability $\mu$ on $(\Bbb R,\mathcal{B}(\Bbb R))$ exists at least a real r.v. $X$ s.t. $P^X=\mu$

Given a probability $\mu$ on $(\Bbb R,\mathcal{B}(\Bbb R))$, does exist always some random real-valued variable $X$ (defined on some probability space $(\Omega,\mathcal{A},P)$) such that its ...
0
votes
1answer
19 views

How bad is batching ECC into distinct cohorts?

The actual problem concerns using redundant error-correcting blocks to protect a collection of data blocks against erasure. Ideally, if I supplied $50\%$ additional blocks, I could tolerate $50\%$ ...
2
votes
1answer
35 views

$\lim_{n \rightarrow \infty} \Bbb P(Y_n >c) =1$ for every $c>0$. Show that $\lim_{n \rightarrow \infty} \Bbb P(X_n+Y_n >c) =1$ or every $c>0$.

I have trouble in a probability problem. Let {$X_n,n \geq1$} and {$Y_n,n \geq1$} be two sequences of random variables such that $\lim_{n \rightarrow \infty}X_n =X$ in distribution for some random ...
0
votes
0answers
17 views

Name for distributions for which all members of the family can be expressed as a transform of a member

Suppose one has a conventionally parameterized Normal Distribution in which the first parameter is a and the second parameter is b. Such a distribution can be expressed as a transform of the base ...
0
votes
1answer
28 views

What is the PDF, CDF, and E[Y] of Y=ln[X+c] if X is lognormal

If $\ln X \sim N(\mu, \sigma^2)$, what is the distribution of $Y=\ln \left(X+c\right)$ where $c$ is a constant. Is this something that can be written out analytically? Also, what is $E[Y]$?
-1
votes
1answer
20 views

How to find the maximum likelihood estimators of parameters in the Pareto distribution? [closed]

Here's the Pareto distribution: $$f(x; \theta_1, \theta_2) = 1 - (\theta_1 /x)^{\theta_2}, \theta_1 \le x, \theta_1, \theta_2 > 0$$ Its likelihood function is complicated and so is its ...