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

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0
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0answers
6 views

Integral of non-Gaussian distribution, random walk?

I would like to evaluate $$ F = \mathbb{E} \left\{ \frac{\int_0^T x^3(t) dt}{\int_0^T x(t) dt} \right\} $$ If $x(t) \sim \mathcal{N}(0, \sigma^2)$ and independent for each $t$, the denominator is a ...
0
votes
2answers
23 views

CDF of sum of N exponentially distributed random variables with condition

I have $Y=X_1u(X_1-x_{th})+X_2u(X_2-x_{th})+\cdots+X_Nu(X_N-x_{th})$, with all the $X_i\sim\lambda e^{-\lambda}$, $u(t)$ is the unit step function and $x_{th}$ being the threshold which means that any ...
0
votes
1answer
6 views

Solving for a Conjugate Prior in search of MAP estimator

I am trying to prove that if a given random variable $X \sim Exp(\lambda)$ and $\lambda \sim Gamma(\alpha,\beta)$ hen $\lambda | X \sim Gamma(\alpha^{*},\beta^{*})$ for some parameters $\alpha^{*}$ ...
-1
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0answers
11 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: ...
1
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1answer
17 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 ...
1
vote
1answer
30 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 ...
1
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0answers
44 views

Limit theorem for changed time

This post seems long, but its almost everything proofed except the last step. The unknown part is marked especially. Given a Levy-Process $U_{t}$ with with $E(U_t)=0$ (then $U_t$ is a martingale). ...
0
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1answer
54 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 $0$ ...
0
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0answers
21 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 ...
0
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1answer
26 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 ...
-3
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1answer
27 views

$P(T ≤ 5 | T ≥ 2)$ from CDF [on hold]

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
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2answers
24 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 ...
0
votes
4answers
36 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 ...
0
votes
1answer
12 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 ...
1
vote
1answer
25 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. ...
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 ...
0
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0answers
14 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 ...
2
votes
2answers
22 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 ...
2
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0answers
25 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 ...
0
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2answers
650 views

Bivariate Normal Distributions

Let X and Y have a bivariate normal distribution with parameters μ1 =3, μ2 = 1, σ1^2 = 16, σ2^2 = 25, and ρ = 3/5 . Determine the following probabilities: (a) P(3 < Y < 8). (b) P(3 < Y < ...
1
vote
1answer
20 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 ...
1
vote
1answer
28 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, ...
0
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0answers
17 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 ...
0
votes
1answer
490 views

Joint probability of sum of two random variables and one of its terms

Let $X$ and $Y$ be two independent random variables (weibull distributed) and $Z=X+Y$. I am trying to find $\mathbb P\big(Z\geq z ~\cap~ X\leq x\big)$. I know that $$ \mathbb P(Z\geq z \cap X\leq x) = ...
-3
votes
0answers
19 views

Distribution of $\frac{\sigma_a^2}{\sigma_2^2+\sigma_e^2}$ [on hold]

Let $\sigma_a^2\sim IG(\beta,1)$ and $\sigma_e^2\sim IG(\alpha,1)$ and take $\rho=\frac{\sigma_a^2}{\sigma^2_a+\sigma_e^2} $. Show that $\rho\sim Beta(\alpha,\beta)$ I don't even get the ...
0
votes
0answers
24 views

Identification of the probability distribution of a discrete random variable and knowledge of its support.

I am confused on the following issue regarding the identification of the probability distribution of a discrete random variable and the knowledge of its support. Let $X$ be a random variable defined ...
0
votes
1answer
12 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
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3answers
7k views

Discrete Probability Problem: determining probability mass function and cumulative distribution function

Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is .4 (a ...
4
votes
2answers
48 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 ...
4
votes
2answers
44 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: $$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\ 0,& \text{otherwise}.\end{cases}$$ ...
3
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0answers
344 views
+50

Create the most 'stressful' tennis game ever!

Some games, such as tennis, use a complicated points system (point, game, set, match; with deuces and tie-breaks) for what would otherwise be an extremely simple and monotonous game. The main reason, ...
1
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1answer
4k views

Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
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0answers
17 views

Poisson distribution - $P(x_1 = k | x_1 + x_2 = n) = \binom{n}{k} \cdot \frac{1}{2^{n}}$.

$X_i \sim Pׂׂ(\lambda)$ I dont know if the events are Independence. I have to prove that $P(x_1 = k | x_1 + x_2 = n) = \binom{n}{k} \cdot \frac{1}{2^{n}}$. My attempt - I know that $P(x_1 = k | x_1 ...
0
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0answers
25 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
0answers
43 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 ...
4
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0answers
56 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 ...
0
votes
1answer
17 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
35 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 ...
-1
votes
2answers
41 views

Probability density function for product and minimum of i.i.d. $U(0,1)$ random variables

If $U$ and $Y$ and $Z$ are i.i.d. $U(0,1)$ random variables, find the pdf for $A= U \times Y$ and $B = \min \{ U,Y,Z\}$.
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0answers
12 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 $(y,\vec{x})$. ...
3
votes
1answer
36 views

Confusion related to calculating the probability distribution of a variable

I have this confusion related to calculating the probability distribution of a variable. If I have a variable $x_1$ which has a pdf $p(x_1)$.Lets assume that the distribution is gaussian with mean ...
1
vote
0answers
7 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 ...
15
votes
1answer
661 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
2
votes
1answer
25 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 ...
4
votes
1answer
53 views

Conditional expectation maximum of sample

Find the conditional expectation $\mathbb{E}\left[\left.X_{1}\right|Y\right]$ if $X_1,..., X_n\sim\mathrm{Uniform}\left(0,1\right)$, where $Y=\max\left\{ X_{1},...,X_{n}\right\}$. MY ATTEMPT: We ...
1
vote
1answer
39 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 ...
1
vote
1answer
1k views

Uniformly Most Powerful Test and Rejection Region of Poisson Distribution

Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$. (1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of ...
3
votes
1answer
49 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: ...
-2
votes
1answer
15 views

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

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 ...
5
votes
2answers
1k views

The logic behind a sequence

I am trying to get the logic behind the sequence: for $n=2,3,\ldots$ $$\left(\frac{\log (2)}{\log \left(\frac{3}{2}\right)},\frac{\log (3)}{\log \left(\frac{17}{9}\right)},\frac{\log (4)}{\log ...