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

learn more… | top users | synonyms

0
votes
0answers
4 views

Divisibility property of Laplace

What does it mean that the laplace distribution has the divisibility property?Is there any distribution that this does not hold?
0
votes
0answers
10 views

Aggregation Urn Distribution

I am trying to identify this distribution in terms of the number of balls, $n$, urns, $m$, and iterations, $i$. Before the first iteration each ball is independent. The first iteration consists of ...
0
votes
1answer
22 views

Conditional Distributions

Choose a random integer $X$ from the interval $[0, 4]$. Then choose a random integer $Y$ from the interval $[0, x]$, where $x$ is the observed value of $X$. Make assumptions about the marginal pmf ...
0
votes
0answers
35 views

How can I find the PDF for this random variable? [closed]

Could you please give me a hint to solve this problems? Assuming $X_i$ and $Y_i$ , $i=1,\ldots,N$ are exponential random variable with mean value $\Omega_X$ and $\Omega_Y$. We define a new random ...
1
vote
1answer
11 views

Sequence of Gamma r.v.s converges in probability to 1

Let $\{X_n\}$ be a squence of Gamma distributed random variables with pdf $$ f(x;\alpha,\beta) = \begin{cases} \hfill \dfrac{x^{\alpha - 1}e^{-x/\beta}}{\beta^{\alpha}\Gamma(\alpha)} ...
0
votes
2answers
41 views

Independent and identically distributed random variables

Let $Y=1/4(X_1 + X_2 + X_3 + X_4)$, where $X_1$, $X_2$, $X_3$ and $X_4$ are i.i.d. r.v.s (independent and identically distributed random variables) with a Cauchy pdf $$f_X(x) = \frac{a}{\pi(x^2 + ...
0
votes
0answers
10 views

deriving frequency distribution from rank ordering distribution of a variable

I have a variable $Y$ whose values can be modelled using the discrete generalized beta function of with rank $r$ of as a parameter. $r$ is rank of a particular value of $Y$. So mathematically it can ...
0
votes
2answers
8 views

Expected delay problem on expectation based on uniform distribution

At a traffic junction, the cycle of traffic light is 2 minutes of green and 3 minutes of red. What is the expected delay in the journey, if one arrives at the junction at a random time uniformly ...
0
votes
3answers
35 views

Finding value of $p(x)$ given an MGF

Question: The MGF of the independent discrete random variable $X$ is given by $$M_X(t) = \left(\frac{1}{2}e^{2t} + \frac{1}{2}e^{4t}\right)^7$$ Find $p_X(15)$ I have been staring at this ...
2
votes
1answer
50 views

Example: convergence in distributions

Give an example $X _n \rightarrow X$ in distribution, $Y _n \rightarrow Y$ in distribution, but $X_n + Y_n$ does not converge to $X+Y$ in distribution. I got a trivial one. $X_n$ is $\mathcal ...
4
votes
0answers
121 views

The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ ...
-1
votes
0answers
14 views

Epsilon Differential privacy using laplace distribution

Given Database X={x_1 ,x_2 ,...X_n} where x_i represent a bit Let f(x) = Sum x_i for 1 <= i <= n Let Y ~laplace(1/(epsilon)) I succeeded proving that Mechanisms: M1(x)= Y-f(x) and M2(x)= ...
-1
votes
1answer
24 views

if a sequence of random variables $X_n \to Y$ in distribution and $X_n \to Z$ in distribution $\Rightarrow$ $F_Y=F_Z$?

In handouts provided by a professor I read: if a sequence of random variables $X_n \to Y$ in distribution and $X_n \to Z$ in distribution $\Rightarrow$ $F_Y=F_Z$. It does not feel right to me. $X ...
0
votes
2answers
25 views

Creating a probability density function from a word problem

I am taking a course related to probabilities and as a primer we are given some word problems, I have somehow slipped by in my earlier classes and never have taken a classes on such subjects. I was ...
0
votes
1answer
17 views

Exponential Distribution Theoretical Quantile

Question Given: For the exponential distribution with CDF: $$F(y) = 1- e^{-\lambda y} $$ show that the (i/n+1)-th theoretical quantile is given by: $$ F^{-1}\bigg(\frac{i}{n+1}\bigg) = ...
2
votes
0answers
40 views

$e^{-d|z|^\alpha}$, $d\geq0,0<\alpha\leq2$, is characteristic function of a stable distribution

Problem: Prove that $e^{-d|z|^\alpha}$ is characteristic function of a stable distribution, if $d\geq0$ and $0<\alpha\leq2$. A note on the definition of stable: Note that a measure $\mu$ ...
-2
votes
0answers
23 views

Probability (Please see picture below) [closed]

On a scratch card you win if you find a sun in the first square you scratch off. Here are the scratch cards before the suns are covered. $$\mathbf{A}\ \ \ \ \ \ \ ...
1
vote
0answers
19 views

Sums of random variables

Good evening community, I need some help. Prove the followed theorems: (a) $X$ and $Y$ have the density functions $p_X$ and $p_Y$ in terms of the Lebesgue-Borel-measure and both are Independent, so ...
0
votes
1answer
27 views

Prove that the integral of Weibull distribution from $-\infty$ to $\infty$ $= 1$

Given: $f(x) = \beta /\theta^\beta x^{\beta -1}e^{-(x/\theta)^\beta}$ this can be rewritten as $$\begin{align} f(x) &= \beta(1/\theta)(1/\theta)^{\beta -1 }x^{\beta -1}e^{-(x/\theta)^\beta} \\ ...
1
vote
0answers
38 views

Expected value and Variance of a stochastic time integral of a deterministic variable (Standard Brownian motion)

Given a Standard Brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$, define: $$E(e^{\int_0^tudB_u})=?$$ $$ Var(e^{\int_0^tudB_u})=?$$ Sidenote to be edited later: Here is my try, I'm not capable to ...
1
vote
2answers
25 views

Bivariate density function

For a distribution function $F$ density function is defined as $\dfrac{\partial^2 F}{\partial x \,\partial y}$. Is it essential that $F$ is differentiable? Is it required that $\dfrac{\partial^2 ...
0
votes
2answers
39 views

Marginal Density Function

Problem: For $y=e^x$ where $f_X(x) = e^{-x}$, $0 \leq x < \infty$, determine $f_Y(y)$. I started to solve this from: $$f_X(x) = \int f(x,y)\,dy \qquad \text{and} \qquad f_Y(y) = \int ...
0
votes
1answer
38 views

Why does the MLE of the uniform distribution not satisfy a Central Limit Theorem?

For $X ~ U(0,\theta$) The MLE of $\theta = \max{x_i}$. Why does this not satisfy $\sqrt{n*I(\theta)} *( \max(x_i) - \theta) -> Z $ Where Z has a normal distribution? I understand that $\max{x_i} ...
-1
votes
0answers
17 views

Rayleigh Quotient

I have two questions: 1) suppose $w \in C^{N}$ is a size-N unit-norm complex random vector, and it is independent and identically distributed with a uniform distribution on the sphere. I would like ...
-3
votes
0answers
57 views

Probability distribution with cumulative distribution function [closed]

Y1, Y2, . . . , Yn form a random sample from a probability distribution with cumulative distribution function FY (y) and probability density function fY (y). Let Y(1) = min {Y1, Y2, . . . , Yn}. 1) ...
2
votes
4answers
96 views

calculating a definite integral of gaussian-like form

As part of a homework question in the course "Introduction to Probability" I take, I was given the following formula: $$\int_0^\infty \exp\left(-x^2-\frac{a^2}{x^2}\right)dx = ...
2
votes
0answers
30 views

Finding the maximum expected value

Say we have $a_1, a_2, ..., a_m \in \{0, 1\}^n$ where the sum over the elements of each vector $a_i$ is $k$. Let $b \in \{0, 1\}^n$ be a random vector based on the uniform distribution. Also, let ...
2
votes
4answers
62 views

Can one generate a sequence of natural numbers whose density has a given distribution?

Suppose $\{ p_{k} \}$ is a collection of real numbers with the following properties: 1) $p_k \in (0,1)$ $~~~~$(i.e. $0$ and $1$ are not allowed values) 2) $\sum_{k=1}^{\infty} p_k =1$ An ...
5
votes
0answers
68 views
+50

Concentration of measure bounds for multivariate Gaussian distributions (fixed)

Let $\gamma_n$ denote the standard Gaussian measure on $\mathbb{R}^n$. It is known (see for example Cor 2.3 here: http://www.math.lsa.umich.edu/~barvinok/total710.pdf) that ...
0
votes
0answers
16 views

Reversing Exponential Distributions (kind of)

I was just wondering how one could go from an exponential distribution to a distribution which describes the probability that a given number of events would happen in the time interval $[0,t]$. They ...
0
votes
2answers
37 views

Probability of matches

The probability of a team winning a match is $\frac{4}{7}$, it losing the match is $\frac{2}{7}$ and it having a draw is $\frac{1}{7}$. If there are 40 matches, then what is the probability that the ...
0
votes
0answers
16 views

constructing various parameter and prediction confidence intervals

We have a correctly specified linear model $y = x\beta + \varepsilon$ with 100 independent observations. Given: $y = e^z$, $\sigma^2 = 4$ (assumed to be known), sample means $\bar{x} = -3$ and ...
1
vote
0answers
14 views

probability integral transformation and distribution of P= P[ |T| <= |t|] .

The task is to find the distribution of P. where , P=P[ |T| <= |t|]. (T is a continuous random variable with PDF f(t)). now , I tried to make the following two arguments : 1.P= P[ |T| <= |t|] ...
1
vote
0answers
8 views

Uniformly minimum variance unbiased estimator

How to prove $ \overline{X}=\frac{1}{n}\sum_{i=1}^nX_i$ is the uniformly minimum variance unbiased estimator of $\mu$ when $X_i\sim N(\mu,\sigma^2),$ and $\sigma$ is known. Idea: Let ...
0
votes
0answers
30 views

Prove that $ P\{ B(n,p) \gt \frac{np}{2} \} \ge \frac{1}{2} $ if $ np \gt 2 $ where B binomial

The question is all in the title. Here $ B(n,p) $ denotes a binomial random variable with parameters n (trials) and p (probability of getting a head). I tried various Chernoff bounds but no one seems ...
0
votes
1answer
30 views

Figuring out the distribution of sample variance

If I have a random sample $X_1,...,X_m$ with normal observations where mean $\mu$ and variance $\sigma^2$ then how can I show that $s_x^2=\sum_{i=1}^{m} ...
1
vote
0answers
17 views

About moments in a quantile processes

Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F: $$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$ In this case, $F^{-1}$ is the (generalized) inverse of ...
1
vote
0answers
20 views

Convergence in probability of the Fisher information

Given a family $\{\mathbb{P}_\theta\}_{\theta\in\Theta}$ on $\mathcal{B}(\mathbb{R})$, where $\Theta\subset\mathbb{R}$ and each member of this family is absolutely continuous w.r.t. $\lambda^1$, and ...
0
votes
0answers
17 views

Distribution - Poisson, Normal and Binomial Question

I have an exam today and can't seem to solve this question: The life time of a chip can be considered according to a normal distribution of mean 6 * 10^4 h and a Standard deviation of 1.5 * 10^4 ...
1
vote
0answers
40 views

Distribution of $(\sup_{0\leq s\leq t} W_s -W_t)$

I am interest in the law of the $(\sup_{0\leq s\leq t} W_s -W_t)$ where $W$ is a standard brownian motion. I know that $M_t:=\sup_{0\leq s\leq t} W_s \overset{\mathcal L}{=} |W_t |$ so its density ...
1
vote
1answer
36 views

Probability distribution of $Y_1=min(X_i)$

Let $Y_1<Y_2<\ldots<Y_n$ denote the order statistics of a random sample of size $n$ from the distribution with pdf : $$f(x;\theta)=e^{-(x-\theta)}I_{(\theta,\infty)}(x)$$ Here we use the ...
0
votes
1answer
33 views

Finding density function

Let Y have a uniform distribution on interval (0,1) Find the density function of U=1-Y^2 I cant graph this, and transformation don't work . Anyone give me a hint on this one? I worked out the ...
2
votes
0answers
17 views

Do pdfs of Logistic and Student's t-distributions intersect only 2 times?

If I have 2 probability density functions of Logistic distribution with the same mean, but different variances, do they intersect only 2 times? Is it also true for Student's t-distribution?
0
votes
0answers
19 views

Looking for a certain probability distribution

I would like to know if one can find a probability distribution with finite mean and the following property: $F(2^{i+1})-F(2^i)\le p$ for given parameter $0<p<1$ all $i$. That is, if we ...
0
votes
1answer
163 views

Expected value of a complicated function of more than one random variable.

Assume we have random variables with Probability Density Functions (pdf) as follows $$\omega_i \sim f_{1},\,\,\,\,\ i \in[1:n]$$ $$ \gamma= \{\gamma_1,\cdots,\gamma_n\} \sim f_2: \text{joint pdf of ...
1
vote
0answers
21 views

How is the proper definition of a continuous density with a point mass?

I would like to define a continuous density function (a.e.) which has also two point masses. For example $$f(x)=0.8\cdot1_{(0,1)}(x)+0.1p_0+0.1p_1$$ where $1$ is the indicator function, $p_0$ is the ...
0
votes
1answer
14 views

If $X$ is distributed exponentially with $\lambda=2$, how can I find $Y=\sqrt{X}$?

I have a random variable $X$, that is distributed exponentially with $\lambda=2$. Therefore it's probability density function is as follows: $$f(x) = \begin{cases} 0, & x \leq0\\ 2\cdot e^{-2x}, ...
0
votes
1answer
21 views

PDF of distance from the center of a random point in the unit disk

if find in a certain website ( also in an IEEE paper) that the probability function for the distance mentioned in the title is given by the following: P(d)=2d, but no one is giving the way to derive ...
1
vote
0answers
49 views

Distribution of an area

I've got the problem of trying to guess the probability distribution of the following area $S$: We draw two lines, $r_1$ and $r_2$, randomly in the plane passing through the origin. So we have two ...
0
votes
1answer
26 views

Dependence of RVs exponentially distributed

Let $X$ and $Y$ be random variables that are both exponentially distributed with different parameters. I dont get the idea how can they be dependent then... and we can only manipulate one parameter in ...