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

learn more… | top users | synonyms

0
votes
0answers
13 views

Relationship between two beta distributions

We have experiment results, which are ratios of poisson variables, of the form X/X+Y, where X and Y are poisson, and X+Y is not fixed. I read that under high lambda, one can approximate these with ...
0
votes
1answer
13 views

How to find Reliability of a rectangular distribution function?

Assume that the failing of a device is equally probable within an interval [a,b] such that the fault density is: f(x) = {1/b-a if a<= t <= b ...
0
votes
0answers
26 views

Joint distribution of random variables

Let $S$ and $T$ two independent random variables. Suppose that S is a standard Gaussian random variable with density $f(s)=(2\pi)^{-1/2}e^{-s^2/2}$ and T with, $2f(t)\mathbf 1_{\{\mathbb ...
0
votes
0answers
16 views

Exact probability distribution for hitting time of simple random walk

Consider simple random walk on the line starting from the site $y \in \mathbb{N}$. With probability $p$ the walker moves to the right and with probability $1-p$ to the left. Call $\tau$ the first time ...
0
votes
0answers
12 views

Multiple probability density transformation [on hold]

I wish to transform the probability density distribution of three independent variables $x_a$, $x_b$ and $x_c$, that appear in the following expression, \begin{equation} ...
0
votes
2answers
27 views

What does “Choose N ~ Poisson(ξ), Choose θ ~ Dir ( α )” mean in the context of Latent Dirichlet Allocation

I'm reading http://machinelearning.wustl.edu/mlpapers/paper_files/BleiNJ03.pdf and trying to understand the notation and concepts behind LDA, in order to implement it myself. I've followed some ...
0
votes
0answers
11 views

Modeling Counts with Small Number of Observations

I have a large data set that contains $5$ different fields. The fields are ...
0
votes
1answer
20 views

limiting joint distribution

Let $X_n\xrightarrow[d]{}N(0,\sigma^2_x)$ and $Y_n\xrightarrow[d]{}N(0,\sigma^2_y)$. $X_n, Y_n$ are not independent. Can I say that $\left( \begin{array} {} X_n \\ Y_n \end{array} ...
0
votes
0answers
11 views

Evacuation schedule length for a linear wireless network?

There is a linear wireless network consisting of node 0, 1, 2, ..., n. Node i can only communicate with node (i - 1) and (i + 1). Each node (i > 0) generates a packet and transfers it to node 0, via ...
1
vote
2answers
42 views

Conditional Expectation Discrete and Continuous

Find $E[X]$ and $Var[X]$ So for the expectation so far I got that: $$E[X] = E[X|N=n]P(N=n) = \large\frac{n+1}{\lambda} \frac{\lambda^{n}}{n!}e^{-\lambda}$$ but for conditioning on both a discrete ...
2
votes
1answer
13 views

Convergence , conditional distribution

here my short question. I saw the following conditional distribution, which converges, in a book: $\lim_{s \to \infty}P\left(\frac{X-f(s)}{g(s)}\leq x\mid X>s\right)=G(x)\ \forall x$ in the set ...
0
votes
0answers
14 views

Implementing the Delta method to assess the confidence and prediction intervals

I want to calculate the table of confidence and prediction intervals for a custom Cumulative Distribution Function or CDF, and I am following the forums and articles aid. My major cuestions that I ...
1
vote
0answers
21 views

Infinite fourth moment and maximum entropy

Alright, I expect this is a silly question, but I don't actually know, so. Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean ...
1
vote
1answer
25 views

Relation between the distribution functions of random variables $Y$ and $-Y$

I'm having trouble understanding a certain property of CDFs for negative random variables. Let $Y$ be an exponential random variable and let $f_y, F_Y$ denote the PDF and CDF respectively. My book ...
0
votes
1answer
24 views

Hypothesis Testing, P-value, T-test Statistic, Confidence Interval

I am writing a report for my class project. I am taking statistics and I am REALLY panicking with the results I have in my report. I do not think my calculations for t-test statistic or confidence ...
0
votes
1answer
26 views

Proving Continuity and equivalence

I have posted ths on the Quant Finance page as it is part of a QF problem but realised I may get a swifter response here! Iam working on a problem where I have successfully reduced a version of Black ...
0
votes
1answer
18 views

Combining biased coin flips

Suppose one has a biased coin $C_1$ with probability $p$ of landing heads and $(1-p)$ prob. of landing tails. If one wants to sample a coin $C_2$ with $p^2$ probability, one can flip the coin $C_1$ ...
1
vote
1answer
28 views

Determine the accuracy of Poisson approximation to birthday problem

I'm currently doing an exploration of the Birthday Problem, and noticed that the formula given to calculate the probability for $m$ people in a room is: $$1-\frac{365!}{365^m (365-m)!}$$ And this ...
0
votes
0answers
41 views

Compound Gaussian distribution

Let $\mathbf{a},\mathbf{b}\sim \mathcal{N}(\mathbf{0},\sigma^2\mathbb{I})$ and let $A$ be the circulant matrix defined to have $\mathbf{a}$ as its first column. I'm trying to study the behaviour of ...
-3
votes
1answer
34 views

Proving independence of variables with normal distribution [on hold]

Random variable $X$ is a variable with standard normal distribution. How to prove that $|X|$ and $\frac{X}{|X|}$ are independent? Thanks.
1
vote
1answer
16 views

Convergence in distribution for changing domains.

I am trying to consider whether this is possible and/or reasonable: Let $X_n:\Delta_n \to \mathbb{R}$ be a sequence of random variables, defined over a unique space $\Delta_n \subseteq \Omega$ for ...
3
votes
0answers
31 views

Probability and sums of prime factors

Of the first N natural numbers, we select two different numbers at random. We'll call the greater one A and the lesser one B. What is the probability (P) that the sum of A's prime factors is LESS than ...
2
votes
1answer
38 views

$X$ ~ $\Gamma(s,\lambda)$. Using $M_X(t)$ find the following…

a) $E(X) =$ ? b) $E(X^2) = $ ? c) $Var(X) = $ ? My thoughts: I know that moment-generating function for $\Gamma(k,\theta) = ( 1 - t\theta)^{-k}$ for $t < \frac{1}{\theta}$. I also know that ...
0
votes
1answer
39 views

Mixing continuous and discrete distributions

I'm wondering how, if it is at all possible, to write the p.d.f. for the following random variable. Given RVs $X_1$ and $X_2$ distributed according to some joint distribution having known density ...
2
votes
1answer
32 views

Deriving a lower bound for a probability involving a random variable $X$ with the Gamma distribution.

Question Let $X$ have the $Gamma(\alpha, \beta)$ density. I.e. $$f_X(x) = \frac{1}{\gamma(\alpha)\beta^\alpha}x^{\alpha-1}e^{-\frac{x}{\beta}}$$ when $x >0$ and $0$ elsewhere. The moment ...
0
votes
2answers
30 views

Probability density use for biased outcome

I am not a mathematics pro so do not mind if this is dumb let us suppose I have a method for generating random real values between 0 and 1 . All the values between 0 and 1 are equally likely to be ...
0
votes
0answers
30 views

Prove Joint distribution of estimators

Let $X_1,...,X_n$ iid r.v. with distribution F, with mean $\mu$ and median $\theta$.Assume that $Var(X_i)=\sigma^2$ and $F'(\theta)>0$. If $\hat{\mu}_n$ is the sample mean, and $\hat{\theta}_n$ the ...
1
vote
1answer
28 views

Probability density function of two uniformly distributed stochastic variables

I'm currently stuck on an exercise involving two independent stochastic variables X and Y. Both X and Y ~ U(0,1) (uniform distribution) The goal of the exercise is to calculate the probability ...
0
votes
1answer
40 views

Simple Expected value of MLE

Let $X_1,..., X_n$ be iid $Exp(\lambda)$. The MLE for $\lambda$ is $\hat{\lambda}=\frac{1}{\bar{X}}$, where $\bar{X}=1/n \sum^n_{i=1}X_i$ How can I conclude that $E(\hat{\lambda}) = n\lambda/(n-1)$? ...
0
votes
1answer
23 views

Distributions of local times of a single excursion of 1D random walk

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
0
votes
2answers
25 views

Finding out the probability distribution of numbers from -3 to 3

I am really able to solve problems with the help of all the volunteers here. a big thanks to every one.. Please explain this problem.. A random variable 'X' takes the values -3,-2,-1,0,1,2,3. ...
1
vote
1answer
16 views

Histogram with different sample probabilities

Assume we are given a list of samples $L_1,L_2,\ldots,L_n$ of some random variable $L$. By classing them into bins we can easily create a standard histogram. But now suppose that we associate a ...
1
vote
0answers
20 views

Working with the sum of two independent random variables, and estimating a parameter

A network source sends a sequence of zeros and ones, $X_1, X_2, ...$ with $X_i$(iid) Bernoulli with $p = P(X_i = 1), 0 < p < 1$. Due to disturbances the received sequence is $Y_1, Y_2, ...$ ...
3
votes
1answer
32 views

Is there a name for the probability distribution with the form $p(x) = a \,x^2\, \exp( -b\,x^2 )$

There is a probability distribution: $$p(x) = a \,x^2\, \exp( -b\,x^2 ), \quad a,b>0,\ x \in ( -\infty,\,\infty ) $$ I wonder which probability distribution is it?
1
vote
1answer
44 views

$E(X_i \cdot I(X_i>\theta)$ expected value of when X is greater than the median.

Let $X_1, ..., X_n$ be iid with a distribution F. Let $\theta$ be the median of F. What is the value of $E(X_i \cdot I(X_j>\theta))$? If $i\neq j$, then $E(X_i \cdot I(X_j>\theta))= 1/2 \cdot ...
2
votes
1answer
39 views

Finding conditional distribution

Let $X$ and $Y$ be independent $Exp(1)$-distributed random variables. Find the conditional distribution of $X$ given that $X + Y = c$ ($c$ is a positive constant). this is my idea: $$f_{X \mid ...
0
votes
2answers
22 views

Laplace transform of noncentral chi-square distribution

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...
0
votes
1answer
18 views

Distribution combinations

How many ways can 25 identical pencils be distributed between two people?.Each all pencils must be shared out. A) Each person must have at least 5 pencils? B) Each person must have at least 7 ...
1
vote
0answers
9 views

elementary proof for discrete Kantorovich-Rubinstein theorem?

For the Kantorovich-Rubinstein theorem, please see the wikipedia page http://en.wikipedia.org/wiki/Wasserstein_metric (which does not contain a reference for the proof). I am only interested in the ...
0
votes
0answers
9 views

Generate quadrature points from a distribution

Is there any method to generate quadrature points from any arbitrary probability distribution, $p_{X}\left(x\right)$? We already know about Gauss Hermite rule for Normal distribution, Gauss-Laguerre ...
1
vote
0answers
33 views

Law and Brownian Bridge

Let $Z_{t}= W_{t}-tW_{1}$ and $Y_{1}=\sup_{0\leq t\leq 1}Z_{t}$, $(W_t, t \geq 0)$ standard Brownian motion Find the law of $Y_{1}$ I know that $\textbf{P}(\sup_{0\leq t\leq 1}W_{t}\geq x , ...
3
votes
2answers
48 views

What is the reason for the one-half in the normal pdf's gaussian (i.e. : why $\exp(-x^{2}/2)$ instead of $\exp(-x^{2})$ )

It doesn't seem to relate to normalization, as the normalizing constant adapts to every possible "upstairs formulation", and in the standard case is $\displaystyle\frac{1}{\sqrt{2\pi}}$. Does it ...
1
vote
0answers
41 views

The probability that exactly / at-least $k$ numbers are in the correct position [duplicate]

Given a sequence of $[1,\dots,n]$ in random order: Let $P_k$ be the probability that exactly $k$ numbers are in the correct position Let $Q_k$ be the probability that at least $k$ numbers are in the ...
0
votes
1answer
44 views

Martingale based on normal PDF evaluated at normalized i.i.d. sums

I have the following problem. $(X_n)_{n\geq0}, n\in\mathrm{R}$, is a family of iid r.v., normally distributed $\mathcal{N}(0,1)$ $\mathcal{F_n} := \sigma((X_i)_{1\leq i\leq n})$ $x\in\mathrm{R}, ...
-3
votes
0answers
9 views

Proof of “Normal approximation to the log-normal distribution” [closed]

I saw the post about the normal approximation to lognormal (Normal approximation to the log-normal distribution). The proof is shown as well. Yet as I'm looking for the proof in a journal article form ...
1
vote
1answer
21 views

Probability with Exp distribution, CDF, and multiple variables

You have a list of chores to do at home, but are expecting family to arrive shortly. The amount of time until their arrival (measured in hours) can be modeled as an Exp(2) random variable. Your list ...
1
vote
2answers
28 views

Probability with Uniform Distribution with Multiple Variables

Every time you go to a beach for vacation, you take home a little sand to keep as a souvenir. Over your lifetime, you have done this exactly 100 times. On each visit, the weight of sand you take home ...
2
votes
1answer
74 views

Probability Question: Who's right, me or the book?

I'll be giving some classes on probability theory later this year, and so I've been going through the textbook to check that I'm up to speed. I came across the following question: The discrete random ...
1
vote
3answers
59 views

Transformation of two independent uniform random variables

Suppose $X,Y \sim \text{Uniform} \left(0,1 \right)$ are independent. Then I need to find the PDF for $W=X/Y$. By the CDF technique this is seen to be : $$F_W( w)=\int_{0}^1 \int_{0}^{wy} ...
1
vote
2answers
34 views

P/1 Actuary Exam Question

I was doing problems and came across this one and was wondering why the P[1<=x<=2] is F(2) - lim (x->1-) F(x) rather than F(2)-F(1)? Could someone please explain this for me?