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

learn more… | top users | synonyms

3
votes
1answer
87 views

Determine the distribution of $\int_0^t (W_s-\frac{s}{t}W_t) ds$, where $(W_s)_{s\geq 0}$ is a brownian motion

I have to find the distribution of $X_t:=\int_0^t (W_s-\frac{s}{t}W_t) ds$ where $(W_s)_{s\geq 0}$ is a brownian motion. I already showed the first integral $\int_0^t W_s ds$ is ...
3
votes
1answer
198 views

Convergence of marginal distribtution

Here I have a question which looks a little bit weird: $(q_n)_n$ is sequence of probability density functions of the couple $(x,y) \in \mathbb R^2$, $p_n$ is the marginal density of $q_n$, i.e. ...
3
votes
1answer
134 views

Exponentially distributed random variables

Given two exponentially distributed random variables $ X_1 $ and $X_2$ (assuming rates $\lambda_1$ and $\lambda_2$ respectively), determine the probability that one is smaller than the other. So ...
3
votes
3answers
1k views

X,Y are independent exponentially distributed then what is the distribution of X/(X+Y)

Been crushing my head with this exercise. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything together. The exercise goes as ...
3
votes
2answers
173 views

Mean of an increasing function over exponential distribution

I came across the following problem in my research I have two random variables $X, Y$ which are exponentially distributed and $Y$ has a higher mean than $X$. Then I have a function, say $f(z)$, ...
3
votes
3answers
324 views

$E[X]$ finite iff $\sum\limits_{n} P(X>an)$ converges

Show that: $$\sum\limits_{n \in N } P(X>an) < \infty\ \text{for some}\ a > 0 \Rightarrow E[X] < \infty \Rightarrow \sum\limits_{n \in N } P(X>an) < \infty\ \text{for every}\ a > ...
3
votes
0answers
160 views

Simplifying covariance matrices in distributions

In the multivariate Gaussian distribution, it is required that the covariance matrix be positive semidefinite. I have read that a positive semidefinite matrix $\Sigma$ can be written as $LL^{T}$. I ...
3
votes
1answer
519 views

Uniform measure on the rationals between 0 and 1

I am trying to think of a probability measure on the set of rationals between 0 and 1 ($X:=\mathbb{Q}\cap[0,1]$). I want to achieve something like a uniform measure, i.e. every number should have the ...
3
votes
3answers
527 views

Expected number of overlaps between intervals

Suppose $N$ intervals of length $\delta$ are positioned in $[0,1]$. The starting point $l_i$ of each interval is drawn from an uniform distribution, i.e., $l_i \in [0, 1-\delta]$, thus it will ...
3
votes
3answers
4k views

The mode of the Poisson Distribution

Lately, I am doing an investigation on Stirling's formula and its applications. So I thought I could use it to prove that the mode of the Poisson model is approximately equal to the mean. Of course, ...
3
votes
3answers
193 views

What is the distribution of $x'Cx$ when $x$ is a standard gaussian vector

When $x$ is an '$n$' dimensional standard Gaussian, we have $x'x \sim \chi^2$ with $n$ degrees of freedom. Now if I have a symmetric matrix $C$, what will be the distribution of $x'Cx$ ? ...
3
votes
1answer
131 views

The limit of the expectation of the top half+1 order stats of $n$ draws of $X$ as $n\to\infty$?

Can anyone help me compute the limit of the average of the top half +1 of order marginal order distribution of $n$ draws from $X$, as $i\to\infty$? Specifically, the limit as $i\to\infty$ of ...
3
votes
2answers
3k views

PDF of product of variables?

could anyone please indicate a general strategy (if there is any) to get the PDF (or CDF) of the product of two random variables, each having known distributions and limits? After having scanned ...
3
votes
2answers
2k views

CDF of a ratio of exponential variables

Let $X$ and $Y$ be independent exponential variables with rates $\alpha$ and $\beta$, respectively. Find the CDF of $X/Y$. I tried out the problem, and wanted to check to see if my answer of: ...
3
votes
2answers
821 views

Random Exponential-like Distribution

Note: Not good at math and my terminology may be very wrong. I have a uniform random number generator that outputs a number between [0,1]. I'd like a function that returns a random number between 0 ...
3
votes
4answers
7k views

Convolution of two Gaussians is a Gaussian

I know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the ...
2
votes
1answer
19 views

Ratio between $k$th highest number among $n$ and $n+1$ samples

Let $n\geq k$ be fixed positive integers, and let $X$ be a distribution on $[0,1]$ that is not the constant $0$ distribution. Let $E_n$ denote the expected value of the $k$th highest value among $n$ ...
2
votes
2answers
92 views

Random sums of iid Uniform random variables

Let $\{X_r : r\ge 1\}$ be independently and uniformly distributed on $[0,1]$. Let $0<x<1$ and define $$N=\min\{n\ge 1 : X_1 + X_2 +\ldots+X_n> x\}$$ Show that $$P(N>n) = ...
2
votes
0answers
26 views

How to calculate the following conditional expectation? Is my calculation process right?

I want to calculate the conditional person's correlation coefficient. But I don't know how to calculate the following expressions,especially the conditional expectation of ...
2
votes
2answers
88 views

Find distribution $Y=X^2$

X~N(0,1). Find distribution $Y=X^2$ Can someone help me? I have no idea how to do it. I could try to start like this: $F_Y(t)=P(X^2<t)=P(-\sqrt(t)<X<\sqrt{t})$
2
votes
1answer
42 views

Closed-form term for this expression

I have a normal Distribution $X \sim N(\mu, \sigma)$. Is there an easy way to give an asymptotic estimate with small error (I would prefer with relative error $\rightarrow 0$) for $P[X \geq k]$? We ...
2
votes
2answers
75 views

From distribution to Measure [duplicate]

I have been asked to create a new post with my question. So it is about starting from a distribution function and proving that we can always find a probability space. My attempt is this : So assume ...
2
votes
1answer
140 views

Which biased random sources can be obtained from an unbiased one?

Let $X_i$ with $i\in\mathbb N$ be a sequence of independent binary random variables with uniform distribution $\operatorname{Pr}(X_i=1)=\operatorname{Pr}(X_i=0)=0.5$. For $p\in[0,1]$ with $p2^n\in ...
2
votes
3answers
88 views

Given $X$ and $Y$ are independent N(0,1) random variables and $Z = \sqrt{X^2+Y^2}$ from the marginal pdf of $Z$

Let $X$ and $Y$ be independent $N(0; 1)$ random variables. Let $Z = \sqrt{X^2+Y^2}$. (a) Derive the marginal pdf of $Z$ and then using the marginal pdf to compute ${\rm E}[Z^2]$ (b) Can you propose ...
2
votes
2answers
330 views

How to show that the difference of two Gumbel distributed random variables follows a Logistic distribution?

How can you show that when you have two random variables $X,Y\sim\text{Gumbel}[0,1]$ , then $X-Y\sim\text{Logistic}[0,1]$ . I tried to use the convolution formula ...
2
votes
2answers
96 views

A basic doubt on joint distribution

How to calculate the following probability $P(X \leq x, Y=y)$ where $X$ is a continuous random variable and $Y$ is a discrete random variable. I have been given the distribution of $X$ and ...
2
votes
2answers
267 views

Integrating a special skew normal — the CDF of a convolution of a normal with a truncated normal

I am having a little trouble trying to compute an integral. In short, I wish to solve the following: $$F(x) = \int_{-\infty}^x \phi(au-b)\,\Phi(au+b)\,du $$ My intuition is that this might be ...
2
votes
2answers
2k views

marginal probability mass functions

Let $X$ and $Y$ be random variables with joint probability mass function $f(x,y) = k \cdot \dfrac {2^{x+y}}{x!y!} $, for $ x, y \in \{ 0, 1, 2, \cdots \} $ and for a positive constant $k$. Derive ...
2
votes
1answer
316 views

Probability of throwing balls into bins

You are throwing n balls into m bins randomly. What is the probability to be empty of the first $k$ bin? Given $k$ bins are empty. What is the probability to be empty of $(k+1)th$ bin? Forget the ...
2
votes
1answer
198 views

Exponential distribution: Finding the parameter

Please help me solve the following problem Time of production of one electronic component is given with exponential distribution with parameter λ. If the process lasts less than 3 hours, the ...
2
votes
1answer
205 views

drawing at least one colored ball of each from urn in a case of large populations

My problem is: If an urn contains balls of $10^7$ different colors, namely $K_1, K_2, \ldots K_{10^7}$, and there are 1000 balls of each color, so that the total number of balls in the urn is ...
2
votes
2answers
141 views

Comparing the relative entropies of some stochastically ordered distributions

Motivation of this question: This question is related to the expected stopping time of a stochastic process under two hypotheses. Especially, it answers the question "how many more samples are ...
2
votes
2answers
78 views

Probability $P(A < B)$

Given two independent and continuous random variables $A$ and $B$ with cumulative distributions $F_A$ and $F_B$, show that $$P(A<B) = \int_{-\infty}^{\infty} F_A(x)\, F'_B(x)\,dx.$$ Is this ...
2
votes
1answer
95 views

distribution function of time T

an ambulance station is located 30 miles from one end of a 100-mile road. the station services accidents along the entire road. suppose that an accident occurs. suppose that Suppose accidents occur ...
2
votes
2answers
434 views

integrating using student t distribution

Evaluate the integral $\int_0^\infty\frac{1}{1+x^2}dx$ using the Student t distribution. I don't know where to start. I am assuming that I can't just do regular integration. I don't know how I am ...
2
votes
1answer
88 views

Behaviour of Two Coupled Sequences Towards a Stable Distribution

The following question arises from research that I am doing in swarm intelligence. The relationships given come from geometric considerations which, I believe, should not be relevant for this problem. ...
2
votes
2answers
1k views

Sufficient Statistic for a Geometric R.V.

I have a problem that I know I am very close to the solution for, but I think I just need some more formatting to make it a really clean proof. The problem goes like this: Suppose X is a discrete ...
2
votes
1answer
99 views

Continuous and non-decreasing but how?

I am reading a paper and the author shows the continuity and monotonicity of a function. It seems so simple to see but I am sorry that I couldnt see the reason. I will be very happy if you can point ...
2
votes
1answer
157 views

Conditional Expectations (Mainly an integral question)

Let $X_1$ and $X_2$ be two Random variables with a standard normal distribution, and the two variables are independent. Find $E[X_1|X_1>X_2]$ My answer is far. If we knew $X_2$, then the answer ...
2
votes
1answer
1k views

Problem deriving Beta distribution normalizing constant

Given beta distribution as: $$ \mathcal{B}(x;a,b) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} x^{a-1} (1-x)^{b-1} $$ I am trying to show: $$ \int_0^1 x^{a-1} (1-x)^{b-1}\,dx = ...
2
votes
0answers
1k views

Expectation of a product of Brownian Motion.

Regarding Brownian Motion formula below, how does $E[W(s)W(t)]$ turn into $$E\left[W(s)\big(W(t)−W(s)\big)+W(s)^2\right]\;??$$ I have asked a question using the formula below, but this and that are ...
2
votes
1answer
3k views

CDF of sum of dependent random variables

Suppose that $X$ and $Y$ are $dependent$ random variables, what would be the cumulative distribution of $X+Y$? That is, what is $P(X+Y\le c)$ for any integer c? Note that we do not know their joint ...
2
votes
2answers
379 views

What is the density of the sum $Z = X+Y$?

Find the density of the sum $Z = X+Y$ when $X$ and $Y$ are independent, standard uniform random variables. $$f_X(x) = 1\quad\mathrm{if}\quad 0\le x \le 1$$ $$f_Y(y) = 1\quad\mathrm{if}\quad 0\le y\le ...
2
votes
0answers
262 views

Determining the probability density function from an equation

I have the following (for me quite interesting) densities for which I am completely stuck. I only hope that you can provide me some help. Let me introduce my problem. I have two probability ...
2
votes
2answers
234 views

Help with the integral for the variance of the sample median of Laplace r.v.

When we draw $n$ samples of Laplace-distributed random variable such that $n=2k+1$ and the location parameter is zero, the median $x$ (or the $k$-th order statistic) has the following p.d.f.: ...
2
votes
1answer
475 views

Find the probability that the second customer to arrive has to wait to be served if arrival time is exponential and serving time is uniform

Customers line up to be serviced according to a Poisson process at an average rate of five per hour. If the time it takes to serve one customer is a continuous uniform random variable on $[0,4]$, ...
2
votes
1answer
312 views

Going from binomial distribution to Poisson distribution

Why does the Poisson distribution $$\!f(k; \lambda)= \Pr(X=k)= \frac{\lambda^k \exp{(-\lambda})}{k!}$$ contain the exponential function $\exp$, while its relation to the binomial distribution would ...
2
votes
3answers
2k views

$3\sigma$ rule for multivariate normal distribution

I was wondering if the $3\sigma$ rule that holds for 1D normal distribution also holds for multivariate normal distribution?
2
votes
3answers
126 views

A probability distribution and random variable

Assume we have $X_1,...,X_n$ independent poisson random variables. What is the cdf or pdf of $ \sum_{i=1}^{n} X_i$ ??
2
votes
1answer
403 views

Combining 1D normal distributions into a 2D distribution

First of all, apologies for my poor terminology - I have a particular problem which I understand in own terms, but I am having difficulty in applying the mathematics in the correct manner. My problem ...