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

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0answers
44 views

Three random variables with exponential distributions

Having $X$, $Y$ and $Z$ as three independent identical random variables all having exponential distribution $E(X)=E(Y)=E(Z)=\frac{1}{\lambda}$, What is the answer of the following probability: $P(X+Y&...
1
vote
1answer
34 views

Why aren't CDFs left-continuous?

Let $F$ be a cumulative density function on $\mathbb{R}$. From an argument in a textbook, it is shown that $F$ must be right-continuous: Let $x$ be a real number and let $y_1$, $y_2$, $\ldots$ be ...
0
votes
0answers
28 views

Why does the following equality hold in proving Meyer's inequality?

I have a question in proving Meyer's inequality. The proof I read is taken from the book "Malliavin Calculus and related topics" by Nualart. I just have one equality which I am not sure, I will ...
4
votes
1answer
148 views

Limit theorem for changed time

This post seems long, but its almost everything proofed in this post. Only one step seems to be left, for the desired proof. I would be very gratefull for any help. The setup Given a Levy-Process $U_{...
0
votes
1answer
29 views

Let $X$ have a Poisson distribution with parameter $\lambda$.

Let $X$ have a Poisson distribution with parameter $\lambda$. (a) Show that the moment-generating function of $$Y = \dfrac{(X − \lambda)}{\sqrt{\lambda}}$$ is given by $$M_Y(t)=exp(\lambda e^{\frac{t}...
-3
votes
1answer
35 views

$P(T ≤ 5 | T ≥ 2)$ from CDF [closed]

If for discrete random variable T the CDF is defined as $$F(t) = \begin{cases} 0, & \text{t<1}\\ 1/4, & \text{1≤t<3}\\ 1/2, & \text{3≤t<5}\\ 3/4, & \text{5≤t<7}\\ 1, &...
0
votes
2answers
31 views

Chebyshev's inequality to find probability of interval

Here is how I solved the problem: $$ X\sim N(\mu=.13, \sigma^2=.005^2)\\ .12\le x\le .14 \\ \mu-2\sigma\le x \le \mu+2\sigma\\ $$ Using Tchebychev's inequality, I get $$ P(|x-\mu|\le 2\sigma)=1-\frac ...
1
vote
1answer
29 views

Can I run a regression when both independent and dependent variables are all dichotomous?

I have conducted a survey where all my questions are asked in a dichotomous manner (Yes/No). Eg IV:"Are you a smoker?", "Are you obese", "Is your gender male/Female" etc. DV: "Have you ever had a ...
0
votes
1answer
14 views

How the value of denominator calculated here?

I found this example in a book and it has to find probability distribution as stated below: If a car agency sells 50% of its inventory of a certain foreign car equipped with side airbags, find a ...
0
votes
4answers
41 views

Convolution: Give a proof that $f_T(t)=\int_{-\infty}^{\infty}f_X(x)f_Y(t-x)dx$ where $f_T(t)$ is the PDF of random variable T

Here is the question: Let $X$ and $Y$ be independent, continuous r.v.s with PDFs $f_X$ and $f_Y$ respectively, and let $T=X+Y$. Find the join PDF of $T$ and $X$, and use this to give a proof that $...
2
votes
2answers
24 views

Getting the marginal distribution from the joint pdf

To bein with, I did the following calculations: $$ Y\sim Uniform(0,x)\\ f_x(x)=\{\frac{1}{x^2},x\ge1\}\\ f_{y|x}(y)=\{\frac{1}{x},0\le y \le x\}\\ f(x,y)=f_x(x)f_{y|x}(y)=\frac{1}{x^3},x\ge 1,0\le y\...
1
vote
1answer
26 views

Joint probability distribution.

I am trying to calculate P(Y|Z) given the following distribution $\ P(X,Y,Z) = P(X)P(Z)P(Y|X,Z)$ Now, initially I did the following calculation. $$P(Y|Z)=\sum_{x}^{}P(X,Y|Z)=\sum_{x}^{}P(X,Y,Z)/P(Z)=\...
0
votes
0answers
23 views

bloom filter: how to estimate probability and tune the filter

My goal is to tune bloom filter in such a way so that I'd get best possible results. I have a dictionary of N=100000 strings, and I have distinct sets of strings S0, S1, S2. For each string from ...
2
votes
0answers
32 views

joint-probability of Langevin equation

I am working on Langevin equations: $\frac{dx}{dt}=u$ $m\frac{du}{dt}= -\gamma u + \theta(t)$ where $\theta(t)$ is delta-correlated in time Gauss-distributed noise with zero-mean $\langle \theta (...
1
vote
1answer
37 views

Change of Uniform Continuous Variable

Let $X$ be a $U(-1, 1)$ random variable, we define $Y = X^4$. Calculate the correlation coefficient between both variables. Are they uncorrelated? PS. I don't know how to use MatJax equations, I'm ...
1
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0answers
24 views

GMM with full and diagonal covariances

I have Gaussian Mixture Model-- distribution with probability density function, that is a weighted sum of Gaussian probability density functions: \begin{equation} p(X)=\sum_{i=1}^k \omega_i\mathcal{N}...
1
vote
1answer
28 views

Distribution of a transform of bivariate to univariate random variable?

Suppose we have two random variables $$R\sim U[1-\varepsilon,1]\;\;\;\;\; \Theta\sim U[0,2\pi],$$ and a third random variable $$X=g(R,\Theta)=R\cos\Theta.$$ What is the density $p_X(x)$? The ...
4
votes
2answers
57 views

Meeting probability of two bankers: uniform distribution puzzle

Two bankers each arrive at the station at some random time between 5PM and 6PM (arrival time for each of them is uniformly distributed). They stay exactly five minutes and then leave. What is the ...
4
votes
2answers
54 views

Prove that $\mathbb P(X>Y) =\frac{b}{a + b}$ if $X, Y$ are exponentially distributed with parameters $a$ and $b$.

Let $X, Y$ be an exponentially distributed random variables with parameters $a, b$. Then $X$ has pdf: $$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\ 0,& \text{otherwise}.\end{cases}$$ ...
0
votes
0answers
19 views

Random variable with characteristic function cosine

So I am searching for a Random Variable $X$, such that $\varphi_X(t)=\cos(t)$. I know how to choose $X$ such that $\varphi_X(t)=e^{it}$ and $\varphi_X(t)=e^{-it}$. Does this help me? How can I put ...
0
votes
1answer
31 views

Using the Central Limit Theorem to calculate a mean from Poisson distributed random variables

Firstly, I am studying the basic concepts of statistics and so any explanations, advice and suggestions are more than appreciated. Onto the problem- I am given the central limit theorem and understand ...
4
votes
3answers
38 views

Birthday line to get ticket in a unique setup

At a movie theater, the whimsical manager announces that a free ticket will be given to the first person in line whose birthday is the same as someone in line who has already bought a ticket. You ...
0
votes
0answers
13 views

What are the differences between stochastic v.s. fixed regressors in linear regression model?

If we have stochastic regressors, we are drawing random pairs $(y_i,\vec{x}_i)$ for a bunch of $i$, the so-called random sample, from a fixed but unknown probabilistic distribution $(y,\vec{x})$. ...
0
votes
1answer
60 views

Let $X$ be a random variable with mean $0$ and finite variance $\sigma^2$. By applying Markov’s inequality show that

I am looking for confirmation that I am working in the correct direction as well as pointers for points where I have gone astray. Here is the problem. (a) Let $X$ be a random variable with mean $0$ ...
1
vote
0answers
12 views

Calculating the normalizing factor in the VonMises-Fisher distribution on $S^p$

I'm going quickly through the VonMises-Fisher distribution $M$ on $S^p$ and its properties. Its probability density function is: $$f(x; \kappa,\mu)= c(\kappa)\exp(\kappa x^T\mu)$$ where $\kappa \...
4
votes
0answers
67 views

Expected length of longest arithmetic sequence

Given a natural number $n$, we define the vector valued random variable $\vec Y_n := (X_1, \ldots X_n)$ where all $X_i$ are independently uniformely distributed on $S_n := \{1, \ldots, n\}$. Further ...
2
votes
1answer
28 views

Find the PDF of $Y= \sin{(\pi X)}$, where $X \sim U[0,1]$

Let $X\sim U_{(0,1)}$ and lets define $Y= \sin{(\pi X)}$. I want to get the pdf of $Y$. My attempt: Clearly, $y\in(-1,1)\Rightarrow 1-y^2\ge0$, so $$ F_Y(y)=\Bbb P(Y\le y)=\Bbb P\big(\sin{(\pi X)}\le ...
0
votes
0answers
30 views

Tail bounds for functions of a Poisson point process

A Poisson point process consists of a sequence of points $0\leq t_1\leq t_2<\cdots$ where $t_i = t_{i-1} + X_i$ where $X_i$ is an exponentially distributed random variable with some rate parameter $...
-2
votes
1answer
16 views

Find the value of $\lim_{n \to \infty}\Pr[\max(X_1,X_2, …,X_n) <a+\ln n ]$ [closed]

Let $X_1,X_2,\dots,X_n$ be independent and $\operatorname{Exp}(1)$ distributed. Calculate the limit $$\lim_{n \to \infty}\Pr[\max(X_1,X_2,\dots,X_n) < a+\ln n].$$ I have tried several things ...
0
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0answers
16 views

calculate the $P(B(1)\leq 0,P(B(2)\leq 0))$, $B(t)$ is the standard brownian motion.

denote $W(1)$ by $(B(2)-B(1))$. then $P(B(1)\leq 0, B(2)\leq 0)$ = $P(B(1)\leq 0, B(1)+(B(2)-B(1))\leq 0)$ =$P(B(1)\leq 0, B(1)+W(1)\leq 0)$ =$P(B(1)\leq 0, W(1)\leq -B(1))$. by conditioning by ...
2
votes
1answer
65 views

What is the pdf of sum of log-normal and normal distribution?

The question goes like this: $Z = X+Y$; where $X$ is Log-normal Random variable with parameters - $\mu = 0 \quad \sigma^2= 1$, $Y$ is Gaussian Random variable with $\mu= 0\quad \sigma^2= 1$ What is ...
0
votes
1answer
35 views

What is the probability of getting exactly one two and one three in a 5 card draw?

In a 52 cards deck, what is the probability of getting exactly one 2 and one 3 if 5 cards are drawn. I'm wondering what is the difference between doing it the following two ways. Intuitively I would ...
1
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0answers
30 views

Probability problem related to Markov inequality

Problem Let $p$ be the probability of a person chosen at random to support Bernie Sanders. A sample is taken of $50$ persons chosen at random, each of them is asked if he or she would vote for ...
3
votes
1answer
52 views

If $\lim\limits_{A \rightarrow \infty} \sup\limits_{n} \frac{\int_{|x|>A}x^2 dF_n(x)}{\int_\mathbb Rx^2 dF_n(x)}=0$ then $\{F_n\}$ is tight

Suppose $X_n$, $n \geq 1$, are random variables with distribution functions $F_n$ satisfying $EX_n^2 < \infty$ for all $n$ and $$\lim_{A \rightarrow \infty} \sup_{n} \frac{\int_{\{x: |x|>A\}}x^2 ...
4
votes
1answer
36 views

Questions about the distribution of $Y$ given the distributions of $X$ and of $Y$ conditionally on $X$

$\newcommand{\Var}{\operatorname{Var}}\newcommand{\E}{\operatorname{E}}$Given: $X$ uniform on $(0,1)$ and $Y\mid X=x$ with distribution $N(x,1)$. Question 1: Determine $\E(Y^2)$ and $\Var (Y)$. ...
6
votes
1answer
233 views

Why was I wrong about the monster-gem riddler

Every week I like to do the fivethirtyeight.com Riddler, an interesting and pleasantly challenging (at least for me) weekly math puzzle which comes out Fridays, with the answer and explanation to the ...
2
votes
1answer
48 views

Probability Mass Function of infinitely re-rolled dice

I play a game called Shadowrun. It is a role-playing game that uses a dice pool mechanic. A player has a dice pool of $x$ six-sided, unbiased dice. Every 5 or 6 counts as a success. The more successes,...
3
votes
4answers
266 views

Condition probability distributions: Two people flipping fair coins

Suppose that two people are playing a game where they each flip a fair coin 100 times. The winner of this game is the person who has flipped the most heads. What is the expected number of heads ...
2
votes
1answer
29 views

Computing the distribution of a uniform r.v. with parameter being another uniform r.v.

I have this: Let $X\sim U(0,1)$, $Y\sim U(X,1)$. What is the distribution of variable $Y$? My answer: I use a geometric approach since everything happens in the square $(0,1)\times (0,1)$, see ...
0
votes
1answer
13 views

Relation between Poisson representation of extremes and GPD representation of extremes

I want to derive the theoretical relation between the parameters in a point process model for extremes and the parameters in the GPD model for extremes. I'm following Coles - An introduction to ...
0
votes
2answers
36 views

On the proof that every positive continuous random variable with the memoryless property is exponentially distributed

The theorem to prove is: $X$ is a positive continuous random variable with the memoryless property, then $X \sim Expo(\lambda)$ for some $\lambda$. The proof is explained in this video, but I will ...
2
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0answers
47 views

How to generate correlated random numbers with specific distributions?

After read the answers of some similar questions on this site, e.g., Generate Correlated Normal Random Variables Generate correlated random numbers precisely I wonder whether such approaches can ...
1
vote
1answer
23 views

Finding the marginal distribution for problem with n balls.

I am trying to solve the following problem: A box contains N balls: $N_1\ white, N_2\ black,\ and\ N_3\ red\ (N = N_1 + N_2 + N_3).$ A random sample of n balls is selected from the box (without ...
0
votes
0answers
18 views

Ratio of Two Sample Mean of Gamma Random Variables.

Let $X_1,\ldots, X_n$ are iid $\mathrm{Gamma}(\alpha,\beta)$, $Y_1,\ldots, Y_n$ are iid $\mathrm{Gamma}(\alpha,\gamma)$ and independent of $X_i$. What will be the distribution of $\frac{\bar X}{\bar Y}...
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0answers
17 views

Is this an exponential family of distributions? from casella and berger 6.20

I am trying to do 6.20 in Casella and Berger part d. The solutions manual says that the order statistics are minimal sufficient and not complete. I understand their logic, but why doesn't this work? ...
0
votes
2answers
33 views

Finding the distribution of a n tossed fair coin

I am trying to solve the problem: Consider a sequence of n tosses of a fair coin. Let X denote the number of heads, and Y denote the number of isolated heads, that come up. (A head is an “...
1
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0answers
47 views

Distributions of components to distribution of vector

Suppose that I have independent variables $x_1,\ldots,x_n$ with tractable (not necessarily identical) distributions. I'm interested in the distribution of $\boldsymbol{x}=(x_1,\ldots,x_n)'$ and, if ...
0
votes
0answers
17 views

Derivation of spacing distribution of independent events

A crude approximation of the spacing of energy levels $E_i$ of complex nuclei (like uranium) is that the energy levels appear independently, with known average spacing $D$. I'm trying to understand a ...
0
votes
0answers
27 views

Reference for function being a distribution function

Let $\mathcal B(\mathbb R)$ denote the Borel $\sigma$-algebra on $\mathbb R$. Let $\nu$ be a probability measure on $(\mathbb R, \mathcal B(\mathbb R))$. Lastly let $F:\mathbb R \rightarrow [0,1]$ ...
0
votes
0answers
15 views

Understanding the posterior of the Dirichlet process

Draws from a Dirichlet process (DP) are discrete, and exhibit clustering behaviour. Suppose I draw $G_{1:5}$ distributions from a DP. Then the posterior probability for $G_6$ is given by (Blackwell ...