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

learn more… | top users | synonyms

2
votes
1answer
76 views

Inequality For Subgaussian Distributions

For my research I am trying to bound some exponential moments of subgaussian r.v.'s. And I am stuck with proving one of such inequalities. More specifically: Let $a$ be unit vector in ...
3
votes
1answer
125 views

Expectation of a compound random variable

Let $X_{1},X_{2},...,X_{n},...$ be independent random variables following the uniform distribution on $[0,1]$. Let $N$ follow the negative binomial distribution with the probability function ...
0
votes
1answer
93 views

Cumulative Distribution Fit

I am looking for a monotonically decreasing function to fit a cumulative distribution. The distribution is the number of values of a random variable X, that are greater than Y as a function of Y. In ...
2
votes
0answers
228 views

Central limit theorem - speed of convergence in center vs tails

I've been told that one of the implications of the central limit theorem is that as we increase the sampling of random variables, we converge faster to a normal distribution in the center and slower ...
2
votes
1answer
53 views

Inferring possible future distribution given observed events.

I have a process which continues for a possibly infinite amount of time, where some event can either happen or not happen, with equal probability (call it $p$) for each time step. I've observed $n$ ...
0
votes
1answer
250 views

If $x = y$ what is $p(x\mid y)$?

I don't speak maths too well (engineer) so simple language preferred or could you describe it as a graph please? This has probably been asked but I have no idea what to search... Follow up question: ...
2
votes
1answer
510 views

Show estimators of P(X=0) for X~POISSON are biased/unbiased

Let $X \sim \operatorname{Poi}(\mu)$ and $\theta = \Pr[X=0] = e^{-\mu}$. Show that $\tilde{\theta} = u(X)$ is an unbiased estimator of $\theta$ where $u(0) = 1, u(x) = 0$, for $x=1,2,3,\ldots$ Is ...
2
votes
1answer
206 views

Product of three Poisson distributions

Product of two Poisson distributions is a Bessel function: $$ \sum_{r=0}^\infty \frac{e^{-f} f^r }{\Gamma(r+1)} \frac{e^{-g} g^r }{\Gamma(r+1)} = e^{-f-g} I_0\left(2 \sqrt{f g} \right) $$ What I ...
3
votes
1answer
339 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 ...
2
votes
1answer
176 views

Problem with coupling (basic probability)

If I have two probability spaces : $\\\Omega_1=\{w^1_1,w^1_2,w^1_3\}$ with $P_1$ defined to be $P_1(w^1_1)=P_1(w^1_2)=P_1(w^1_3) = 1/3$ and $\Omega_2=\{w^2_1,w^2_2,w^2_3\}$ with $P_2$ defined to be ...
1
vote
1answer
163 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
0
votes
1answer
2k views

Likelihood function of a gamma distributed sample

I missed the day of class where we went over likelihood functions, and I had a quick question. If $X_1,...X_n$ are i.i.d. $ {\Gamma}(\alpha,\beta)$ r.v.s, I'm trying to find the likelihood function ...
2
votes
1answer
579 views

Negative Binomial Question Without Exact Values

The question I am working on is: Three brothers and their wives decide to have children until each family has two female children. What is the pmf $X=$ the total number of of male children born to ...
1
vote
0answers
2k views

Finding the mean and variance of the sum of three exponentially distributed random variables

Given: $A = X + Y + Z$, the parameter of $Y$ and $Z$ is $\mu$, the parameter of $X$ is $\lambda$. The coefficient of correlation of Y and Z is $\beta$, $X$ and $Y$ and $X$ and $Z$ are pairwise ...
0
votes
1answer
178 views

Probability question involving two independent poisson random variables?

A restaurant has two waiters. suppose that the number of customers serviced daily by each waiter can be viewed as independent Poisson random variables with parameters $\lambda$ and $\mu$. whats the ...
4
votes
0answers
146 views

Using Bernoulli distribution approximate the $q$-th moment

Let $x$ be vector in $R^n$. Let $\pi(⋅)$ be a permutation on the set $\{1,\ldots,n\}$ with a uniform distribution. Let $|m|\leq n, m \in Z$. Using Bernoulli (or maybe some other) distribution ...
2
votes
0answers
17 views

3 Component Series System

An engineering system consisting of three components is configured as a series system. The components are acting independently of each other with the ith component's lifetime $T_{i}$ having an ...
0
votes
1answer
49 views

(Probably difficult) inequality question for densities under different meaures

Given the measures $(G_0,G_1)$ and $(G^{'}_0,G^{'}_1)$ corresponding to the densities $(g_0,g_1)$ and $(g^{'}_0,g^{'}_1)$ the following inequality $$G_0[g_1/g_0<t^{'}]\geq ...
2
votes
0answers
139 views

What is the total variation measure of the integration of a kernel of signed measures?

Assume given a probability space $(\Omega,\mathcal{F},P)$ and a measurable space $(E,\mathcal{E})$. Let $(\nu_\omega)_{\omega\in\Omega}$ be a family of signed measures on $(E,\mathcal{E})$. Assume ...
0
votes
1answer
498 views

Simplifying a seemingly complex probability problem.

We randomly put $100$ numbered balls in $100$ baskets, and if I am to ask what's the probability of the third ball being in a basket between the first and second ball, I know the answer is exactly one ...
0
votes
1answer
104 views

Help with Probability distributions

You are conducting a study of the relationship between the amount of rain in a field and the total mass of fruit produced by tomato plants. You randomly select tomato plants from a field and weigh ...
0
votes
1answer
92 views

Find the limiting distribution of the following trinomial distribution

Let $X_1,X_2,\dots$ be i.i.d. with the following probability density: $$P(X_j=0)=1-\frac{\lambda}{n}$$ $$P(X_j=1)=P(X_j=2)=\frac{\lambda}{2n}$$ Define $Y_n=\sum_1^nX_j$. Find ...
1
vote
1answer
75 views

About the differential entropies of well-known continuous distributions

Assume that the continuous random variable $X$ has a distribution (in a closed form expression) with differential entropy $h(X)$. Q) Then, is it true for any continuous distribution that the ...
10
votes
3answers
4k views

Expectation of the min of two independent random variables?

How do you compute the minimum of two independent random variables in the general case ? In the particular case there would be two uniforms variables with difference support, how should one proceed ? ...
1
vote
1answer
695 views

Find the probability mass function of the (discrete) random variable $X = Int(nU) + 1$. [duplicate]

For a non-negative real number $x$, write $Int(x)$ for the largest integer that is less than or equal to $x$. Let $U$ be a uniform random variable on $(0,1)$ and $n \geq 1$ an integer. Find the ...
4
votes
0answers
240 views

An absolutely continuous cumulative distribution function that fails to have a Riemann-integrable pdf.

We know that if a real-valued random variable $ X $ on a probability space has an absolutely continuous cumulative distribution function (cdf) $ F $, then $ X $ possesses a probability density ...
1
vote
3answers
178 views

What is the $\operatorname{cov}(X, \max(X,Y))$ and $\operatorname{cov}(X, \min(X,Y))$ where $X,Y \sim N(0,1)$?

having trouble with this one. The exact questions is the $\operatorname{cov}(X, \max(X,Y))$ and $\operatorname{cov}(X, \min(X,Y))$ where $X,Y \sim N(0,1)$. i think the way to calculate it is to get ...
1
vote
3answers
141 views

Statistics - Cumulative Distribution Function

$$\begin{equation}F(x)=\begin{cases}0 &\quad x<-10 \\ 0.25 & \quad -10\leqslant x <30 \\ 0.75 &\quad 30 \leqslant x <50 \\ 1 &\quad 50 \leqslant x ...
2
votes
1answer
493 views

Linear Combination of Exponential Random Variables [duplicate]

Let $Y \sim \exp(\delta)$ and $T \sim \exp(\lambda)$, and $Y$ and $T$ are independent. How do I get the density $f(x)$ where $X=Y-cT$, $c>0$? Thanks.
0
votes
2answers
162 views

Modified monty hall problem

Hello how to show the following You are given the choice of 3 doors. Behind one is a car and the other two are goats. You pick a door uniformly at random say 1, and Monty opens another door, say 3 ...
-2
votes
4answers
1k views

Find the expected value of $\frac{1}{X+1}$ where $X$ is binomial

The problem: X is a binomial random variable, find $E[\frac{1}{X+1}]$ n and p are not given PDF for a binomial distribution is $\binom{n}{k}p^k(1-p)^{n-k}$ Expected value is $\sum{x_ip(x_i)}$ But ...
1
vote
2answers
71 views

Distribution question

(i) Let $X\mid Y \sim \text{Poisson}(Y)$, and $Y \sim \text{Exp}(\lambda)$. Find the distribution of $X$. (ii) Let $X\mid Y \sim \text{Poisson}(Y)$, and $Y \sim \text{Poisson}(\mu)$. Show that ...
2
votes
1answer
51 views

Find the distribution - Exp and Geom

Let $T = X_1 + X_2 + ...+ X_N$, where $X_i \sim (iid) Exp(\lambda) $ and $N \sim Geom(p)$, such that $P(N=k) = p(1 - p)^{k-1}, k=1, 2, ....,$ and $N$ and $X_i$ are independent $\forall i$. Find the ...
1
vote
2answers
1k views

Poisson process. Time between two events.

Suppose that people immigrate to a territory according to a Poisson process with a $\lambda =$ rate of 1 per day. What is the probability that the time between the tenth and eleventh exceeds two ...
1
vote
1answer
103 views

Help with understanding the $\chi^2$-distribution

I'm studying statistics and there's one part in my book I can't understand. I tried to make as good translation as I can of the problematic part...here goes: Chi squared $\chi^2$ distribution Let ...
1
vote
3answers
111 views

Poisson process. How to solve?

Suppose that people immigrate to a territory according to a Poisson process with a $\lambda =$ rate of 1 per day. What is the expected time until the tenth immigrant arrives?
0
votes
1answer
49 views

What is the frequency distribution of a random variable representing the result of the launch of one dice?

What is the frequency distribution of a random variable representing the result of the launch of one die? I don't if that is correct but: $S = \{1,2,3,4,5,6\}$ The probability of $\{1,2,3,4,5,6\}$ ...
1
vote
2answers
101 views

Calculate a CDF given two PDFs.

I have two PDF's, $f_1(x)$ and $f_2(x)$ and I need to find the CDF (not PDF) of $f_3(x)$ where $f_3(x) = \frac{1}{2} (f_1(x) + f_2(x))$ I have already calculated $F_1$ and $ F_2$, the CDF's of ...
1
vote
2answers
91 views

Uniform distribution - derive joint expectations

Let $X$ and $Y$ be independent random variables with uniform distribution on $[0,1]$, in notation: $X$~$Unif(0,1)$, and $Y$~$Unif(0,1)$. Derive (a) $E(min(X,Y))$, (b) $E(|X - Y|)$, (c) ...
0
votes
1answer
242 views

Maxwell-Boltzmann velocity PDF to CDF

I need to draw from a Maxwell-Boltzmann velocity distribution to initialise a molecular dynamics simulation. I have the PDF but I'm having difficulty finding the correct CDF so that I can make random ...
0
votes
0answers
55 views

Deriving the process of successfully consumed requests from the process of request-producers and the process of request-consumers

The title is not very straightforward I understand, but you will soon realize it was not so simple to describe in few words this problem. The problem Consider a system consisting of: A process of ...
0
votes
1answer
272 views

Wiener process with a random mean [closed]

I have found this kind of stochastic process $$ dX=dW-{\rm sgn}(dW)dt. $$ What would the probability distribution be for $X$ assuming that the distribution for ${\rm sgn}(dW)$ is a Bernoulli with ...
0
votes
1answer
46 views

Cumulative Function to Density Function

Simple Question: How can I find a Density Function of a variable from a Cumulative Function? Example: Cumulative Function: $$F(x) = \begin{cases}0 & x < 1 \\ x^2 & 1\leq ...
0
votes
1answer
111 views
2
votes
1answer
95 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 ...
1
vote
0answers
104 views

Halloween candies!

Children go trick-or-treating in three mathematicians' apartments. In MathA's apartment, a child will roll a die and the number of candies the child receives will be the same as the outcome of the ...
1
vote
0answers
132 views

Probability, Poisson Process, My solution is correct?

Jobs are submitted for a computer according to a Poisson process with a rate λ (jobs / hour). Determine the probability of at least two jobs are submitted within the 'a' first few minutes. I tried ...
1
vote
0answers
24 views

Showing that a distribution can be sampled from 2 uniforms.

Hi there I am looking at the following: Show that the symmetric triangular distribution, $$\begin{equation} g(x)=\begin{cases} 0, & \text{if $x<0$}.\\ x, & \text{$0\leq ...
0
votes
2answers
47 views

Probability, Random points in rectangle

There is a rectangle, the lower left is always fixed at co-ordinate $(0, 0)$. Let the width and height of the rectangle be $w$ and $h$. Let $P$ be a randomly chosen point from the rectangle with ...
1
vote
1answer
385 views

Sufficiency and UMVUE for Poisson distribution

I need to show that $\hat\lambda = \bar X$ is a sufficient estimator for a Poisson distribution iid $X_1...X_n$, show that $\hat\lambda$ is the UMVUE for $\lambda$ and that $\hat\lambda$ is a ...