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

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4
votes
2answers
65 views

CDF of probablity distribution with replacement

I want to get every color of gumball in a gumball machine (where there are 16 types of gumballs, each with a 1/16 chance of obtaining a particular color [assume there are an infinite amount of ...
4
votes
1answer
86 views

Slowly varying function without limit at infinity

A function $f:\mathbb R \to \mathbb R$ is slowly varying at infinity if for any $t>0$ $$ \lim_{x\to +\infty}\frac{f(xt)}{f(x)}=1. $$ Is there a bounded function slowly varying at infinity whose ...
4
votes
1answer
448 views

Maximum order statistic for Binomial distribution

Let $X_i$, $1\le i\le t$, be $t$ independent random variables with Binomial distribution $B(n,\frac1t)$. I would like to find the distribution of $X_{Max}=\max_{i=1}^t(X_i)$ Note that this is the ...
4
votes
2answers
165 views

Verifying Exponential Family

Why is the following $$f(x|\theta) = \theta^{-1} \exp(1-(x/\theta)), \ \ 0 < \theta < x <\infty$$ not an exponential family? We know that $$f(x| \theta) = h(x)c(\theta) \exp(w(\theta)t(x))$$ ...
4
votes
3answers
564 views

Top 3 of 4 Dice Rolls

I'm trying to prove why the mean of the distribution of sums of the top 3 out of 4 fair 6 sided dice is rolls 12.25. Anybody who's rolled a D&D character knows the idea. $r_n = Rand([1,6])$ $x ...
4
votes
3answers
289 views

Probability distribution function

I am trying to develop a function that will allow me to input a random number between 0 and 1 and receive a value. The idea is that the function has a range (for example, 0-100) with a median value of ...
4
votes
3answers
2k views

calculate the Probability density function of the absolute difference of two random variable

If $X$ and $Y$ are two independent random variables with probability density functions $f$ and $g$, respectively, then the probability density of the difference $Y − X$ is given by the ...
3
votes
0answers
33 views

Distribution and convergence of the r.vs.: $X_n= \frac{ \lfloor nX \rfloor}{n}$

$X$ is an absolutely continous random variable, with a continous density function, and: $$X_n= \frac{ \lfloor nX \rfloor}{n}$$ What is the distribution of $X_n$, and what can we say about its ...
3
votes
1answer
89 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
199 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
136 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
175 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
332 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
529 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
535 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
838 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
93 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
27 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
141 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
2answers
344 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
98 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
272 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
321 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
205 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
206 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
1answer
216 views

Conditions for positive dependence

Consider two random variables $X$ and $Y$ with joint distribution $F_{X,Y}$ and strictly positive density function $f_{X,Y}$. Additionally, let $x^*$ be the value of $x$ that solves: $$ \Pr[Y\leq ...
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
445 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
2answers
904 views

Difference of two discrete random variables

Given two random variables $X$ and $Z$ on non negative integers. Supposing that $Z$ is a Poisson distribution of Poisson constant $\lambda$. $X\le Z$ and that $\forall n\ge0,\ \forall k\le n,\ ...
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 ...