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

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14 views

Even moments of distribution given probability density function

Given the probability density function $f(x)$, and the $𝔼[X] = \frac{2}{\sqrt{\pi\lambda}} $, how best should I go about deducing the even moments of this distribution? $f(x) = ...
-1
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1answer
35 views

How to find the Probability of $X \gt 1$ for $p(X)=2e^{(-2X)}$ [on hold]

A PDF is given by formula $p(X)=2e^{(-2X)}$, $x>0$. Determine P($X \gt 1$)? we input 2 and got $=.03663$ but we do know as $X$ goes to infinity $p(X)$ will be $2$.
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0answers
18 views

Distribution of convex combination of Bernoulli random variables

Suppose $Y_1,Y_2, \ldots $ are i.i.d Bernoulli$(p)$. What is the distribution of $$\sum_{i=1}^{\infty} \frac{Y_i}{2^i}$$ I could deal case for $p=\frac{1}{2}$ using characteristic functions but for ...
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0answers
7 views

how to calculate cumulative distribution function inverted exponential in this pic

how to calculate cumulative distribution function inverted exponential in this pic enter image description here
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3answers
21 views

Binomial Random Walk

For the random walk with step sizes: $S_i = \begin{cases} &+1 &\text{probability} &p, \\ &-2 &\text{probability} &q=1-p \end{cases}$ Let $T_n = \sum_{i=1}^mS_i$ be the ...
2
votes
1answer
56 views

Die that never rolls the same number consecutively

Suppose we have a "magic" die $[1-6]$ that never rolls the same number consecutively. That means you will never find the same number repeated in a row. Now let's suppose that we roll this die $1000$ ...
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2answers
18 views

A coin is tossed 6 times. What is the probability that the no. of heads in the first 3 throws

A coin is tossed 6 times. What is the probability that the no. of heads in the first 3 throws is the same as the number number in the last three throws? To be honest, I don't know how to tackle this ...
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1answer
18 views

Distribution of $aXa^T$ for normal distributed vector $a$

Let $a$ be $1\times n$ random vector with entries chosen independently from normal distribution with zero mean and unit variance. What is the distribution of $aXa^T$ for a given $n\times n$ matrix ...
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0answers
8 views

Intuitive difference between Laplace functional of Poisson Point Process (PPP) and independently marked PPP

The Laplace functional of the Poisson Point Process (PPP) $\Phi$ with intensity measure $\Lambda$ on $\mathbb{R}^d$ for non-negative function $f(x)$ is: $$ \mathcal{L}_\Phi(f) = ...
2
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0answers
38 views

Distribution of $1/U^2$ where $U$ is uniformly distributed on $(-1, 1)$

Suppose $U\sim \mathrm{Uniform}(-1, 1)$. Let $Y =1/{U^2}$. What is the distribution of Y? Here is what I have: $$ \begin{aligned} Y \in [1,\infty)\\ P(Y <y) = P\Big(\dfrac{1}{U^2} < y ...
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1answer
33 views

Help in probability, Difficult Question:// [on hold]

Upon testing 80 resistors manufactured by a certain company, it is found that 15 resistors failed to meet the tolerance design specifications a) Construct a 92% two-sided confidence interval for ...
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1answer
11 views

Distribution of inner product of normal random variable by a vector

Suppose, that we have a random vector $\mathbf{x} \sim \mathcal{N}(\mu,\Sigma)$. What is the distribution of $(a\cdot x)$, where $a$ is a real vector? It is known, that for a nonsingular real matrix ...
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4answers
272 views

Central Limit Theorem and Mean time between failures

I was reading up about RAID, and the text said: Suppose that the mean time to failure of a single disk is $100000$ hours. Then the mean time to failure of some disk in an array of 100 disks ...
1
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0answers
8 views

Integral of product of two inverse distribution functions

I am trying to evaluate the integral $$\int_0^1F_{Y}^{-1}(1-u)F_{Z}^{-1}(1-u)\,du$$ where $F_{Y}^{-1}$ and $F_{Z}^{-1}$ are two generalized inverse cumulative distribution functions. What I mean is ...
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1answer
32 views

Finding PDF of function of a random variable

Suppose $X$ has PDF: $f_X (x)= \lambda e^{-\lambda(x+2)}$ , for $x \ge-2$ $f_X(x)=0$ , for $x <-2$ Determine the PDF of $Y = X^2$. I am stuck because for $-2\le X \le 2$, $0\le Y \le 4$, and I ...
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0answers
15 views

Set interpretation - topology vs probability

Consider the sequence of i.i.d. distributed random variables $(X_i)_{i\geq1}$ on $\mathbb{Z}^d$. We define the following norm $I(x)=\mid x\Gamma^{-1}x\mid$, where $\Gamma$ denote the covariance matrix ...
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0answers
88 views

How to prove these two random vectors has the same distribution?

I find this problem when reading a paper. The author seemed to think it is trivial so did not list it as a lemma or something. The question is : $\tilde{u}$ is a random unit vector in $R^D$, $u$ is a ...
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0answers
17 views

The distribution of the condition number for complex Wishart matrices.

I am trying to derive the distribution of the condition number for centered uncorrelated complex Wishart matrices $n\times n$ with $m$ degrees of freedom. The problem is with the solution I got (it's ...
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0answers
16 views

Multivariate normal distribution problem

Consider three Gaussian variables $X_1,X_2,X_3$ with $\mathbb{E}[X_i]=0$ and $\mathbb{E}[X_iX_j]=\rho_{ij}$ for $i,j=1,2,3$. Then, three new variables are defined: $$ \left\{ \begin{array}{l1} ...
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2answers
21 views

Hazard rate question (Exercise 4.4.7 from Grimmett and Stirzaker)

Exercise 4.4.7 asks for the hazard rate of $X$ where $X$ has the Weibull distribution: $$ P(X > x) = e^{-\alpha x^{\beta - 1}}{\rm \hspace{1cm} where\ } x \geq 0. $$ I computed the answer to be ...
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0answers
8 views

distribution of a NIG process under Esscher transform

$Z^{NIG}$ is a negative inverse Gaussian process with parameters $(\mu, \delta, \alpha, \beta)$. We have a $\vartheta \in \mathbb{R}$ such that $E[\exp(\vartheta Z_1^{NIG})]<\infty$. We know this ...
2
votes
3answers
69 views

$X,Y$ are iid from distribution $F$, which is a continuous function, then is $P(X=Y)>0$?

Suppose $X,Y$ are iid random variables from a distribution function $F$, which is a continuous function. Then is it always true that $P(X=Y)=0$? For me, the answer is trivially YES. We have $\int_y ...
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1answer
30 views

When will all the flowers blossom?

The title is not actually correct, but I chose appeal over correctness ;) I'd like to model a flower blossoming cycle, and these are the assumptions: 1) The instant $T$ in which each flower starts ...
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0answers
14 views

Survival and Cumulative distribution of Binomial [on hold]

Could someone please point me to a document that presents the Survival and Cumulative distribution of the Binomial distribution? Even better, could you provide the equations directly?
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0answers
13 views

How to do non-analog sampling from a pdf

I have a distribution given by $$ f(x)=\frac{1}{2}\sinh(\sqrt{2x})e^{-x}. $$ Due to the shape of this distribution, most of my samples will be in the range $x\in(0,2]$. However, I am interested ...
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0answers
25 views

Expected maximum of maxima

Let $F(x)$ denote some CDF, and $\{f_i\}_{i=0}^m$ be a set of random variables independently drawn from that distribution. I would like to compute $$ E\bigg[ \max\bigg\{ \max\bigg\{\{f_i\}_{i=0}^m ...
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0answers
6 views

Distribution of an autoregressive process

Say that we are given a AR process. Also, lets assume that the residuals of the process come form a distribution $P_R$ which, while known to us, is not necessarily normal. Can I derive the ...
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votes
1answer
36 views

Absolute value of a random variable

I have never encountered this concept before. Is this equation valid for $y>0$? $$\mathbb{P}(|X|>y) = \mathbb{P}(-|X|<y<|X|)$$ What about this? $$\mathbb{P}(|X|>y) = ...
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1answer
41 views

Differences of heads and tails in a fair coin

I'm very new to this so I would appreciate a detailed explanation. I wrote a very simple Matlab program that "flips a coin" (randi([1 2])) $n$ times. Every time I ...
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0answers
8 views

Which probability distribution(s) $f(x)$ allow for a closed form solution to $\int\left(x-a\right)^{-\gamma}f\left(x\right)dx$?

I'm trying to find if there is a specific probability distribution $f\left(x\right)$ (or many) such that the following integral $$\int\left(x-a\right)^{-\gamma}f\left(x\right)dx$$ has a closed form ...
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0answers
11 views

How to approach deriving a folder distribution's pdf from original pdf?

Let's say I have an Erlang-distributed random variable $x$, and now I'm only taking the samples of $x$ for which it holds that $x>T$, where $T$ is some constant. The probability $P[x>T]$ can be ...
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1answer
41 views

If $X = X_1+\dotsb+X_N$, and $N\sim\operatorname{Pois}(\lambda)$, then what is the distribution of $X$ given $N$?

I have a question that I'm really struggling with (below): It's hard to understand exactly what is the question actually states. does this mean the number of trials itself is a distribution with a ...
1
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1answer
68 views

Sum of probabilities is infinite

I'm stucked solving this problem: Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with exponential distribution and $\lambda=1$. Show that ...
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0answers
4 views

Existence of first passage time density for time-inhomogeneous diffusion

Let $X$ be a time-inhomogeneous diffusion process in $\mathbb{R}^d$: $$dX_t=b(t,X_t)dt+\Sigma(t,X_t)dB_t,$$ where $\Sigma_{d\times d}$ is uniformly elliptic, and coefficients are such that the above ...
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1answer
56 views

I am not given figures to answer this question. Whats the right approach?

Z is a random variable defined as the sum of N independent Bernoulli trials where the probability of each Bernoulli taking the value 1 is given by p. The number of Bernoulli trials N is itself a ...
2
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0answers
25 views

Distributions on the simplex with beta marginals

Consider a random vector $(X_1,X_2,\ldots,X_n)$ such that 1) $\; X_i\sim\text{Beta}(a_i,\sum_{j\neq i}a_j)\qquad i=1,2,\ldots,n$, 2) $\; X_1+X_2+\ldots+X_n=1$. Can we conclude that ...
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2answers
41 views

Probability problem with combination of poisson and binomial distributions

Exercise The number of clients that enter to a bank is a Poisson process of parameter $\lambda>0$ persons per hour. Each client has probability $p$ of being a man and $1-p$ of being a woman. After ...
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0answers
11 views

Is output from softmax function continuous probability distribution?

I want to ask: Is a output from softmax function a continuous probability distribution?
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0answers
17 views

measure-theoretic probability, (sets of) null events and non-zero probability

Assuming a well-defined probability space $ (\Bbb{R},\mathscr{B},\Bbb{P}_X) $, where $\mathscr{B}$ is the Borel $\sigma$-field, and for a random variable $X$ having a continuous probability density ...
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1answer
49 views

Infinite probability density?

I've read that for a "[..]random variable strongly "localized" around a single value", the probability density function (PDF) could be: $p(x)=\frac {1}{2\epsilon}$, with $\epsilon \to 0$, and ...
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0answers
27 views

Waiting Time Distribution

Let X be a random variable which denotes the amount of time spent in a state(say state 'I') before changing state. As X is a random variable it must have a Probability space/sample space and a sigma ...
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0answers
11 views

Security of $(k, 2k)$-bit generator for small seeds

Here is the problem I am working on for context. I have $\epsilon \le 1 - 2^{-k}$ and $\epsilon$ approaches 1 as $k \to \infty$ but I'm stuck on part c). The $f$ is secure iff there does not exist ...
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1answer
28 views

square-root rule of time

I tried to test the square-root-rule of time for quantiles of a normal distribution. So i created with the statiscal programming language R two variables a<-rnorm(100,mean=2,sd=1) ...
2
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1answer
25 views

probability problem with Poisson distribution

Problem A retailer knows that the demand of boxes is a random variable with Poisson distribution of parameter $\lambda=2$ boxes per week. The retailer completes his stock on monday so as to have four ...
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0answers
18 views

Change of variable formula for density function

We all are aware of the change of variable formula whereby if $$[A, B] = g(X, Y) $$ and g is invertible, then the joint density function of A, B is given by $$f_{ab} (A, B) =1/|J| f_{XY} (g^{-1}(a, ...
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1answer
25 views

Finding distribution of random variable

During my exam there was the following question which I could not answer: Let $X_1, X_2$ be real valued random variables. Assume that $X_1$ is exponentially distributed. Given that ...
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0answers
22 views

show that $U_n$ converges to $0$ in $L^1$ and almost surely.

let $(X_n)_{n\geq1}$ be a sequence of independent random variables. Suppose that the density function of $X_n$ is: $$ f(x)=\dfrac{1}{2}.e^{|x|} \quad x \in \mathbb{R} \quad \forall n \quad ...
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1answer
41 views

Expected value of exponential function

Suppose two identical component are connected in a piece of factory equipment. The two lifetimes X1 and X2 are independent each having exponential distribution with beta =2. The value of the equipment ...
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0answers
19 views

Function of two continuous random variables. find CDF [closed]

[\begin{array}{l}{\rm{Let X be a continuous random variable with uniform distribution on }}\left[ {0,1} \right].{\rm{ }}\\{\rm{Let Y be a continuous random variable with uniform distribution on ...
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1answer
39 views

Let X be a continuous random variable with pdf… [on hold]

a.) Let X be a continuous random variable with pdf $f_x(t) = \exp[-t-e^{-t}]$ for all t in the reals. Find $F_X(x)$ My solution is $$F_X(x)= P(X \le x) = ...