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

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1answer
27 views

Calculate the mean, the median and the quartiles.

Let $D=\{(x,y):x>0,x^2+y^2<1\}$ and let $(X,Y)$ be the random variable with the density: $$f(x,y)=\frac{2}{\pi}1_{D}(x,y).$$ Let $Z=\frac{Y}{X}$. Calculate the mean, the median and the first and ...
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0answers
7 views

Obtaining the Log-logistic distribution from a truncated logistic distribution

Let $$f(x) = \frac{e^x}{(1+e^x)^2}~,~ -\infty \lt x \lt \infty~~~~~(1)$$ be the standard logistic pdf of a random variable $X$. Then one can obtain the pdf of the log-logistic distribution via the ...
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1answer
25 views

Calculate the probability given by three random variables

Let $X_1,X_2,X_3$ be IID random variables, each with the density $$f(x)=x e^{-x}\cdot 1_{(0,\infty)}(x).$$ Calculate $P(X_1+X_2+X_3>4,X_1+X_2<4)$.
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2answers
21 views

Convergence in law of sample means of random variable

Let $\{X_n | n \in \mathbb{N} \}$ be a sequence of independent identically distributed random variables with density function: $$f_X(x) = e^{\theta - x}I_{(\theta, \infty)}(x)$$ with $\theta > ...
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0answers
14 views

Show that for any geometric random variable $X$ and parameter $p, \mathrm{Pr}(X < t) = 1 − p^t$. [on hold]

How to prove the above stated equation? I tried the following : Pr⁡(X(i=1)^(t-1)▒〖Pr⁡(X=i)〗 =∑(i=1)^(t-1)▒〖p(1-p)i-1〗 =1-(1-p)t-1
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0answers
31 views

ODE for the normal distribution [on hold]

The normal density function $\phi(x)=\tfrac{1}{2\pi}e^{-\frac{x^2}2}$ can be described via the ODE $$\phi^\prime(x) = -x \phi(x)$$ under the condition $\int_{-\infty}^\infty \phi(x) = 1$. Is there ...
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0answers
26 views

How to demonstrate the pdf of $P_{\sigma} (t)=\lambda_c e^{- \lambda_c t} / (1 - e^{- \lambda_c T})$

In $t_c$, there are $n$ expirations of $T$ and the remnant $\sigma$ seen from the above figure. Let the time $t_c$ forms the exponential distribution with parameter $\lambda_c$. How to demonstrate ...
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0answers
21 views

Find the moment generation function of $Y=1-e^{-X}$. [on hold]

If $X$ is random variable with PDF: $f(x)=e^{-x}$, $x>0$. Then find the moment generating function of $Y=1-e^{-X}$. Okay, so I don't get why $Y$ is equated to small $x$ since $Y$ itself is a ...
1
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1answer
35 views

A Seemingly Trivial but Computationally Complicated Probability Problem

Suppose $X,Y$ are independent $Uniform(-1,1)$ random variables. Determine the distribution of $Z=X-Y$. I do not really think I should add my work here because whatever I have tried until now, has ...
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0answers
7 views

Sufficient condition for not having infinitely small modes in a distribution

I was reading the paper Optimal Throughput and Delay in Delay-tolerant Networks with Ballistic Mobility (http://dl.acm.org/citation.cfm?id=2500432), and found the following proposition (page 305): ...
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2answers
41 views

Help me understand how to take derivative of the PDF of X~binom(n,p) with respect to p.

This is the solution I was given. My questions: Why is it summed from k=1 to x. Shouldn't it be from k=1 to n? (If not, why not?) What is happening to the first term from line 1 to line 2? When we ...
0
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2answers
25 views

How can I calculte the probability of $X$ with a Generlized Hyperbolic Distribution?

I would like to know how to calculate the probability of $X$ when I have fitted a Generalized Hyperbolic Distribution to my data set. The depth of my knowledge is basic t-tests and z-tests. I am ...
0
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1answer
27 views

Comparing sums of random variables

Consider $X_0,X_1\ldots,X_n$ mutually independent and $X_i \sim U(a_i,b_i)$. What is the probability that $\sum_{i=1}^n X_i<X_0$? Can you extend to mutually independent random variables with ...
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0answers
7 views

Approximation of objective based on statistical distance

I am a computer science researcher (mostly theoretical) currently in midst of statistics and not able to figure out how to proceed. At an abstract level, I have a hypothesis for an unknown ...
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2answers
78 views

Comparing uniform random variables.

$X$ is a uniformly distributed random variable on $(0,1)$ $Y$ is a uniformly distributed random variable on $(0,2)$ $Z$ is a uniformly distributed random variable on $(0,4)$ What is the probability ...
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0answers
35 views

$E_n =\lbrace X_n > X_m \ \forall m < n \rbrace $ are independent

I'm stuck with this exercise. Suppose $(X_n)$ are independent random variables defined on $(\Omega, \mathfrak{F}, P)$ with the same p.d.f. Let $E_1 = \Omega$ and for $n \geq 2$ $$E_n =\lbrace X_n ...
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3answers
37 views

Probability of number of people in car park at any given time

A building has 22 car spaces, each having a car parked within each spot in the morning. Each car is retrieved by its respective owner at some point (random time) between 7am and 9am (120minutes). Each ...
0
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1answer
27 views

Expected valued of Random sums about dice and jar problem

A six-sided die is rolled , and the number N on the uppermost face is recorded. From a Jar containing 10 tag numbered 1,2,,,,10 , we then select N tags at random without replacement. Let X be the ...
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2answers
24 views

Conditional probability about card picking.

A card is picked at random from N cards labeled 1,2,3,,,,,N and the number that appears is X. A second card is picked at random from cards numbered 1,2,3,,,X and its number is Y. I am asked to ...
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0answers
25 views

How to calculate probability of users generating distributed events reaching n events per 15 minutes?

We have games & apps that connect to services such as Facebook and Twitter to fetch information. These services have various rate-limit caps that you cannot exceed - typically based on a 15 minute ...
1
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1answer
18 views

Independence of two multivariate normals.

Suppose we have two multivariate normals $X_1 \sim N(u_1, \Sigma_{11}\Sigma_{22}$) and $X_2 \sim N(u_2, \Sigma_{21} \Sigma_{22})$ . Why are $X_2 $ and $X_1-\Sigma_{12} \Sigma_{22}^{-1}X_2$ ...
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2answers
51 views

Deriving a joint cdf from a joint pdf

I see that a similar question was asked last year, but I am still confused. I have $f(x,y) = 2e^{-x-y}$, $ 0 < x < y < \infty $ and need to find the joint CDF. I have a solution that ...
0
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0answers
15 views

Y(t) = X(t + d) - X(t), where X(t) is a gaussian stochastic process. [on hold]

Could anyone please help me with these questions: a) Calculate the PDF of Y(t) b) Calculate the joint PDF of Y(t) and Y(t + s) I know that if X(t) was iid it would be much easier to be solved. ...
1
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1answer
35 views

Showing That Two Normal-Based Random Variables Have the Same Distribution

Above is my question. $\overline X$ has distribution $N(0,1/n)$ - that's fine to work out. Similarly, $X_n / \sqrt{n}$ has distribution $N(0,1/n)$. These follow from the general relation $$ ...
1
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1answer
19 views

Find the probability generating function of $2X$.

If $X$ follows a poisson distribution with parameter $\lambda$ (mean). Then find the probability generating function of $2X$. I'm getting stuck with forming the expression, as I'm getting confused ...
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votes
2answers
14 views

Let $X$ be a Random Variable. Define $2X$.

I would like to know what exactly the changes are in the values the random variable($2X$) can take, if for example $X$ follows a Poisson or Binomial Distribution. If suppose $X$ follows a Poisson ...
1
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1answer
18 views

Why does a process only satisfy the Markov property if and only if the random times are exponentially distributed?

Given, for example, a birth death process with a set of jump times. These jump times have to be exponentially distributed in order for this process to satisfy the Markov property. Why is this? Why ...
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4answers
69 views

Difference between $E[X^2]$ and $E[X^3]$

Hope to ask a dumb question. $Y = aX$,with $a \in N_+$. Here, we know the correlation coefficient is 1. Now, suppose $X \sim N(0,1)$. Here, we know $X, Y$ are not independent. Cov($X,Y$) = ...
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0answers
12 views

To sketch a “typical” plot of a specific time series model

Let X have a distribution with mean $\mu$ and variance $\sigma^2$, and let $Y_t = X$ for all t. Sketch a “typical” time plot of $Y_t$. My thoughts: This process $Y_t$ is stationary with mean $\mu$, ...
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0answers
8 views

Just like Box Muller algorithm for random numbers in Gaussian distribution, are there any such algorithms for other distributions?

I want to create random numbers in various distributions like Poisson, Binomial, Gamma, etc. I cam across Box-Muller algorithm for random number generation in Gaussian distribution. Are there similar ...
0
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0answers
36 views

Let $X_1,X_2\sim N(0,1)$. How to find joint pdf of $\,Y_1=X_1^2+X_2^2\,$ and$\,\,Y_2=\frac{\displaystyle X_1}{\displaystyle \sqrt{X_1^2+X_2^2}}$?

Let $X_1,X_2\sim N(0,1)$. How to find joint pdf of $\,Y_1=X_1^2+X_2^2\,$ and$\,\,Y_2=\frac{\displaystyle X_1}{\displaystyle \sqrt{X_1^2+X_2^2}}$? $$$$ I have tried to use Jacobian matrix to do ...
1
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0answers
20 views

How to compute the covariance matrix of a random variable uniformly distributed in an ellipsoid

Suppose that x is a random variable uniformly distributed in an ellipsoid \begin{equation} x^{T}Mx\leq\delta, \end{equation} where $x\in \mathbb{R}^{n}$. Clearly, the mean of $x$ is zero. The ...
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0answers
9 views

On Conditional distribution of the multivariate normal.

Following the answer to this question. Where we are talking about a multivariate normal than has mean and covariance matrix that can be decomposed as: $\boldsymbol\mu = \begin{bmatrix} ...
0
votes
1answer
57 views

Integral $\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$

How to evaluate: $$\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$$ Basically am evaluating value of $\log(1+c\chi^2)$ where $\chi^2$ is $\chi$-squared distributed
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0answers
14 views

A simple question about Delta Method's demonstration.

Suppose that $\sqrt{n}(X_n-\mu)\stackrel{D}{\longrightarrow}X$ and consider $g:\mathbb{R}\rightarrow\mathbb{R}$ a function such that first derivative $g'$ is continuous in a neighbourhood of $\mu$, ...
1
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1answer
20 views

Find and sample minimum of two exponential distribtions

I have two (or more) independent exponential variables $ X_1 \sim \exp(\lambda_1) $ and $ X_2 \sim \exp(\lambda_2) $. I want to get both the value of $ \min(X_1, X_2) $ and $ \arg\min(X_1, X_2) $. Can ...
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0answers
11 views

Calculation of probabilities in Z table

I would like to calculate at least one probability from z table. I know that pdf for N(0,1) is 1/(2*pi)*exp^(-(x^2/2)). Also the cdf is However, I do not know how to calculate this integral. ...
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1answer
15 views

explanation of probability density function

How can we explain that if a random variable $X$ has pdf $f(x)$ then the function $Y=g(X)$ will have different pdf than $f(x)$ ?? And how to find the pdf of $Y=g(X)$ ??
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1answer
20 views

sum of two dependent random variables

Let $X$ be a cotinuous random variable uniformly distributed over $[-10,10]$. Let $Y$ be a random variable with pdf $f_Y(y) = \frac{1}{40}\ln \frac{20}{|y|}, -20 \leq y \leq 20$. $X$ and $Y$ ARE NOT ...
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1answer
49 views

Do not exist IID random variables $X, Y$ such that $X-Y \sim U[-1,1]$

This is an exercise from Williams, Proability with martingales. Prove that if $Z$ has the $U[-1,1]$ distribution, then $$\phi_Z(t) = \frac{\sin t}{t}$$ Then prove that do not exist IID random ...
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0answers
8 views

transformation and functions of random variables

Let $X,Y$ be independent random variables. I already have the distribution of $XY$ over a certain subinterval of $\mathbb{R}$, by convolution. My question is, is it possible to get the distribution of ...
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2answers
16 views

Value of lambda in poisson distribution

I am currently studying statistical estimators and I came across a question that asks to give an estimate of the parameter λ of a Poisson distribution (using the method of moments), given that the ...
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1answer
25 views

probability of X+Y which are two independent random variable & uniform distribution[0,1] [duplicate]

Two random variables X, Y are independent and both uniform-distributed in[0, 1]. How to calculate the probability density function Z=X+Y ? I tried below, $$f_X(x) = \begin{cases} \frac1{1-0} \\ ...
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0answers
36 views

Probability distributions with closed-form cumulative distribution functions (CDFs)

I am interested in finding multivariate probability distributions for which the cumulative distribution functions (CDFs) are given in close form. For instance, the multivariate Gaussian distribution ...
1
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1answer
29 views

Summation of binomial number of poisson random variables

Z is summation of K random variables that each has Poisson distribution with different means. But, K is a Binomial random with parameters of n and p. I was wondering what is the distribution of Z?
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2answers
34 views

If x has a distribution function $F_x(x)$, what is the distribution function of $y = \exp(x)$?

I'm really struggling to figure out this problem from one of my practice exercises for a probability course. I know that the probability distribution function $f_x(x)$ is related to the cumulative ...
1
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0answers
16 views

Probability Formula for Posterior With 3 Variables

First post on math.stackexchange; pardon me if this is naive/a repeat. I'm following this document here by Prof. David M. Blei: ...
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1answer
25 views

Distribution of random variables when combined

I need help with this problem: If $X$ and $Y$ are two independent random variables and are both standard normal, what is the distribution of $\frac{1}{2}(X^2+Y^2)$? I think I start with ...
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1answer
24 views

How do I prove that a given probability distribution is Gaussian

I am trying to plot the distribution of a random variable $x$. I got this distribution by marginalising a wishart distribution. When I plot the distribution curve of $x$, it looks like bell shaped ...
0
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1answer
25 views

Find conditional probability of random variables

I need to find conditional probability to count mutual information. Random variable X has uniform distribution on set ...