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

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1answer
158 views

Poisson's distribution , Frustration solitaire

I m not sure how to solve this question from Introduction to Probability & statistics for scientists and engineers,Sheldon M. Ross. Need help. The game of frustration solitaire is played by ...
0
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1answer
34 views

What would this distribution be?

If I flip a coin $n$ times, count the heads, put an $x$ over the number of heads, then I repeat this with $n-1$ heads, and repeat what function would these $x$'s aproximate, I think it might be $f(x)=...
0
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0answers
28 views

Does $x^p/(1+x)$ where $x \in (0,1)$ and $p > 1$ represent any known distribution?

I apologize for this seemingly stupid question. I was working through integrals involving Normal and Cauchy distributed random variables, and I came across this form: $ f(x) \propto x^p/(1+x) , x \...
1
vote
1answer
124 views

How To prove Any Change to $v=a\cdot y + b$ maks $y=(a)^{-1}(v-b)$ Uni. random value

This question may seem to be related to Probability and Data Integrity but mine is much simpler and consideres a DIFFERENT problem. Let a finite field be $\mathbb{Z}_p$, where $p$ is a prime number. ...
0
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1answer
35 views

Random Variable which takes two distributions with different probabilities

Suppose $X$ is a random variable which is $~N(0,1)$ with probability $0.4$ and $~N(2,1)$ with probability $0.6$. What is the pdf of $X$ ? Is it gaussian ? My attempt : Let $f(.)$ represent the pdf ...
2
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2answers
51 views

Infinite sequence of exponentially distributed random variables

Consider an infinite sequence of exponentially distributed random variables, $X_k$, where$ k \in \{1, \ldots, n\}$ with $\lambda = 1$. I am trying to evaluate: $$\lim_{n\to\infty} \frac{\max_{1 \leq ...
4
votes
2answers
2k views

Poisson Distribution when only given using mean

I'm doing the following homework problem and am unsure of whether or not my answers are correct. This is my first time working with Poisson distribution and I want to make sure I am doing it correctly....
2
votes
1answer
126 views

Buffon Laplace Needle Problem

With a and b positive numbers, a needle of length $l\in(0,min(a,b))$ is dropped randomly on a rectangular grid consisting of an infinite number of parallel lines distance a apart, and perpendicular to ...
0
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1answer
72 views

Change of variable in a probability density function

In Bishop's Pattern Recognition and Machine Learning, I'm confused by the section on the change of variable of a probability density function. The paragraph concerned is bellow. So, given a change ...
4
votes
2answers
69 views

Joint distribution of $X_1/(X_1+X_2)$ and $X_2/(X_1+X_2)$ for independent exponential random variables $X_1$ and $X_2$

If $X_i$, $i=1,2$, are independent $gamma(\alpha_i,1)$ random variables. Find the joint distribution of $X_1/(X_1+X_2)$ and $X_2/(X_1+X_2)$. I am trying to use transformation method to solve it. Let ...
1
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1answer
40 views

Jointly distributed random variable $P(X^2+Y^2) \lt 1$

I am analyzing the below problem: Let $X$ and $Y$ be random variables with joint pdf $$f_{X,Y}(x,y)=\begin{cases} \frac{1}{4} & -1 \leq x ; & y \leq 1 \\[6pt] 0 & \text{otherwise} \...
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1answer
28 views

Conditional expectation and its multiplication

In the ross's book, Stochastic Processes, he asserts the following argument. Let ${f_{\tau}}_n(x)=\frac{n}{t} (\frac{x}{t})^{n-1}, \quad 0<x<t$. Then $E[\frac{1}{\tau_n}|\tau_n>y] \cdot P\{...
-1
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1answer
40 views

How To Prove that $v=r\cdot z\ $ is Distributed Uniformly at Random [duplicate]

I consider a finite field $\mathbb{F}_q$, where $q=2p+1$, and $p$ , $q$ are prime numbers. Let $z$ be a fixed element of the field. Also let $r$ be a value picked uniformly random from the field ...
0
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1answer
35 views

Is $v=\ (r_i)^{-1}\cdot z $, a uniformly random value of a field?

We consider a finite field $\mathbb{F}_q$ where $q=2p+1$ and $q$ and $p$ are prime numbers. Let $r_i$ be a value picked uniformly at random from the field such that $r_i>\frac{q}{2}$. Let $z$ be a ...
5
votes
1answer
122 views

Max and sum of random variables

I have a set of independent random variables $\{A_1, A_2, B_1, B_2\}$. All of them have the same distribution function $F(x)$. I want to find distribution function of a variable $C$, where $C=max(A_1 +...
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0answers
63 views

Probability distributions for complex numbers

Aside from the complex normal distribution (which is really just a bivariate normal applied to the real and imaginary parts), are there any other distributions involving complex numbers that have ...
-1
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1answer
45 views

Probability and Data Integrity

This question is about probability and Security (i.e. data integrity). The scenario I am going to explain is a client-server case where the server may modify the client's data. We define a field $\...
1
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0answers
47 views

Which multivariate Gaussian has the highest expected norm?

Let $X$ be an $n$-dimensional Gaussian vector with zero mean and covariance matrix $K$ given by: $$K_{ij} = \begin{cases} p_i(1-p_i) & i=j \\ -p_ip_j & i\neq j\end{cases}~,$$ where $(p_i)_{i=...
0
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0answers
28 views

Finding the Complicated Posterior Probability Distribution of $θ$

Suppose, we are given a likelihood function, $f(x|θ)$ which follows a shifted-exponential distribution and the prior distribution of “$θ$” is Standard Cauchy distribution. Now the problem is – I am ...
0
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1answer
89 views

Is $P( (A\cap B)\cap C)$ equal to $P(A\cap C) P(B\cap C)$?

Is $P( (A\cap B)\cap C)$ equal to $P(A\cap C) P(B\cap C)$? In a proof I doubtfully used this equation. Is it correct? But I am not sure about it. Can somebody confirm its validity? If possible, can ...
1
vote
1answer
52 views

Conditional expectation of iid nonnegative random variables

I am studying Ross's book, stochastic processes. There is the following lemma: Let $Y_1, Y_2, ... , Y_n$ be iid nonnegative random variable. Then, $E[Y_1+ \cdots +Y_k | Y_1+\cdots+Y_n=y] = \frac{...
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2answers
53 views

Why distribution of multiple recursive random number generators is uniform?

I was reading the article of L'Ecuyer on random number generation. The title of this article is "Uniform Random Number Generation". One of the proposed PRNGs there, is multiple recursive random ...
0
votes
1answer
65 views

Monotone likelihood property and first order stochastic dominance

I have a question regarding first order stochastic dominance. Give two pdf $f(x)$, $g(x)$, $x\in[x_0,x_1]$. For all $x$ on the support, I have $$ g(x) = f(x)\cdot H(x) $$ where $H(x)$ is continuous, ...
2
votes
2answers
63 views

Random walk in one dimension with different probabilities

As the title suggests, I'm concerned with a typical random walk problem, where the probability to go right is $p$ and the probability to go left is $q=1-p$. I was trying to find the probability of ...
2
votes
1answer
40 views

Distribution function technique - check my approach?

so I'm learning a new topic and am a bit new to probability, so please excuse my elementary question. Given a random sample on an exponential distribution with mean $\theta$ of $X_1,X_2,...,X_n$, let ...
2
votes
1answer
312 views

transformation of uniformly distributed random variable f(x)=1/2pi into Y=cosx

Let $X$ be a uniformly distributed function over $[-\pi􀀀;\pi]$. That is $ f(x)=\left\{\begin{matrix} \frac{1}{2 \pi} & -\pi\leq x\leq \pi \\ 0 & otherwise \end{matrix}\right.\\ $ Find ...
2
votes
1answer
408 views

Explicit examples of smooth entropy computation

Smooth classic entropies generalize the standard notions of entropy. This smoothing stands for a minimization/maximization over all events $\Omega$ such that $p(\Omega)\geq 1-\varepsilon$ for a given $...
0
votes
1answer
39 views

Poisson sampling

Suppose I have a pdf $f(S)$. $f(S)$ describes the size of firms in the economy. Also define the Bernoulli variable $X_{f} \in \{0,1\}$ where $P(X_{f}=1)=g(S_{f})$ and $P(X_{f}=0)=1-g(S_{f})$. $S_{f}$ ...
0
votes
2answers
55 views

T distribution with n degrees degrees of freedom

I would like to prove that $\displaystyle \frac{\bar{X}\,\sqrt{n}\,}{\hat σ^2}\sim t_{n}$. Note that x~N(0,$σ^2$) and they are iid. Could someone explain why $\displaystyle \frac{\bar{X}\,\sqrt{...
12
votes
1answer
105 views

Distribution of $\sum\limits_{i=1}^{N}X_{i}$ conditionally on $\sum\limits_{i=1}^{N}X_{i}^{2}$ for i.i.d. standard normal $X_i$s

Assume that the random variables $X_{i}$ are i.i.d $\mathcal{N}\left(0,1\right)$, then: $$S_N=\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)\qquad\qquad T_N=\sum_{i=1}^{N}X_{i}^{2}\sim\chi^{2}\...
1
vote
1answer
34 views

Random variable with density function that is scaled geometric mean of density functions of two independent normally distributed random variables

Given two independent normally distributed random variables A and B: $$A \sim \mathcal{N(\mu_A, \Sigma_A)}$$$$B \sim \mathcal{N(\mu_B, \Sigma_B)}$$ is there a way to find normally distributed random ...
0
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1answer
30 views

Understanding function's notation

I have been given a question on the following pdf: Suppose the random variable, X, follows a uniform distribution on the interval (0, θ). The pdf of X is $f(x;θ)$ = $1/θ$, $if$ $0≤x≤θ$, $θ>0$, $0$...
0
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1answer
14 views

Walking through the reduction of a cumulative probability function to a polynomial

Setup Define $P(p)$ as follows: $$ P(p) = \sum_{N_1-\phi \cdot N_2 \geq \theta} {n_1 \choose N_1} {n_2 \choose N_2} p^{N_1 + N_2}q^{n_1 + n_2 - N_1 - N_2}. $$ Here, $$ q = 1 - p. $$ The sum is ...
0
votes
2answers
74 views

PDF of $Y=\min(0,X)$ when PDF of $X$ is $\frac34(1-x^2)$ on $(-1,1)$

Let $X$ be a random variable with density $f(x) = (3/4) (1-x^2).$ Range is $-1 < x < 1.$ I have to find probability distribution of $Y = \min(0,X).$ I know that distribution function could be ...
1
vote
1answer
68 views

Change of variable using dirac delta function

How do I intuitively understand the following result to find the probability density function $P_Y(y)$ given $P_X(x)$ after change of variables $y=f(x)$ or several variables. How to derive this from ...
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0answers
22 views

Conditions for Mellin inversion

Under which conditions is the function $$ g(s)=a^{c(s-1)}\Gamma(s),\qquad a>0,c\in \mathbb{R} $$ the Mellin transform of a probability density function $f$? If $c=-1$, then $f$ is the exponential ...
1
vote
1answer
45 views

apply the law of total expectation

I'm a little bit confused about applying the law of total expectation. Suppose $v_1,v_2,v_3$ are three random variables drawn independently from the same distribution $\mathrm{uniform}(0,1)$, which ...
1
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2answers
33 views

Name of the probability distribution

If $X\sim N(0,1)$, then the density function of random variable $X^3$ is as follows: $$f(y)=\frac{1}{3\sqrt{2\pi}}\left | y \right |^{-\frac{2}{3}}e^{-\frac{1}{2}\left | y \right |^{\frac{2}{3}}}$$ I'...
0
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1answer
49 views

Combined arrival rate

Let us suppose a scenario with two clients, $a$ and $b$, each one generating load at rate $\lambda_a$ and $\lambda_b$, respectively. The server receives the requests from both clients. What will be ...
2
votes
2answers
797 views

Generalized chi distribution

Let $v\in\mathbb{R}^n$ follow a multivariate Gaussian$(0,I)$ distribution, and $M\in\mathbb{R}^{n\times n}$ a matrix. Has the distribution of the Euclidean norm $\|Mv\|$ been studied? I know that its ...
0
votes
0answers
48 views

Probability Density following affine transformation

Suppose $X$ is a random variable in $R^n$ and $Y=a^{T}X+b ∊ R$. If $f_X$ is the density of $X$, then what (and how!) can I obtain $f_Y$ the density of $Y$? It is assumed that $a\neq 0$. I saw the ...
2
votes
3answers
99 views

Adding two discrete distributions

I am taking a probability course and I am having trouble adding two discrete distributions. The two distributions given are: $X$ has a discrete uniform distribution on the integers $0,1, ... ,9$. $...
3
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0answers
38 views

Wireless networks on two sequential office floors: Random partitions of a finite interval via a point process on a line

Construct a Poisson point process of density one on a line of length $L$. Allow each point in the process to "see" part of the line to their left, and part of the line the their right (such that the ...
1
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1answer
16 views

Finding out the percentage points.( F - Distribution).

How to find the values of these $x_1$ and $x_2$ , given , $P(x_1<F_{7,7}<x_2) = 0.90$ , using the F-Distribution tables.. Can anyone provide me a hint for this ?
0
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1answer
20 views

Derivation of t(n-1) distribution

While trying to prove that $\displaystyle \frac{\bar{X}\,-\,\mu}{S/\sqrt{n}}\sim t_{n-1}$ I came across a manipulation that I can not seem to understand the reasoning behind it. Why does $$\frac{\...
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0answers
83 views

Polynomial Interpolation When part of $y_i$'s are Shuffled

Hypothesis: Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and ...
4
votes
1answer
386 views

“General” non centered Chi distribution (having correlated random variables)?

Let $\mathbf{X} = [X_0, X_1]^t \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ with $\boldsymbol{\mu} = [\mu_0, \mu_1]^t \in \mathbb{R}^2$ and $ \boldsymbol{\Sigma} = \begin{bmatrix} \...
1
vote
1answer
192 views

Conditional probability distribution with geometric random variables

Let X and Y are independent random variables following geometric distribution with parameter p. Find the distribution of X given that X + Y = n. I made it this expression... $$P\{X =i|X+Y=n\}=\frac{(...
1
vote
1answer
58 views

Writing the expected value of a random variable in terms of its cumulative distribution function

My professor said that an alternative expression for the expected value of a random variable can be written as: $$ E[X] = \int_{0}^{\infty} (1-F_X(x)) \, dx - \int_{-\infty}^0 F_X(x) \, dx $$ No ...