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

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0
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3answers
26 views

Find C and the distribution function

The density distribution of $\xi$ is defined by: $P_{\xi}(x)= \begin{cases} Cx^{-3/2} & \text{if } 1\leq x, \\ 0 & \text{if } x<1 \end{cases}$ Find C and the distribution function ...
2
votes
1answer
24 views

What model should I use for judging a dimension given only composed data with another?

I am attempting to upgrade a modeling system using a limited type of statistical information, but with the sample covering the entire system. The problem is how to use the additional information in ...
0
votes
1answer
25 views

Generating Normals with specific means and variances

Suppose I wish to generate normals $X, Y, Z$ with the correlation matrix R but with means $0, 1, 2$, and variances $4, 16, 25$, respectively. How would you do this? The only way I know of doing ...
1
vote
1answer
42 views

Related problem to covering a circle with random arcs

I have a problem setup wherein we have (the following are all integers) a sequence of length $G$, and $N$ reads of length $L$. I'm interested in the problem where we consider the sequence to be ...
0
votes
3answers
69 views

$Y$ can only take on $\{−1, 0, 1\}$. The expected value of $Y$ is $0$ and its variance is $1/2$. Find the probability distribution of $Y$.

How would one approach this question? A random variable Y can only take values in $\{−1, 0, 1\}$. The expected value of $Y$ is $0$ and its variance is $1/2$. Find the probability distribution of $Y$. ...
2
votes
1answer
19 views

Max Likelihood Examples, Stuck in Calculation [closed]

We get samples 2,4,8,16 be random instances that get from distribution with following PDF. maximum likelihood estimation of $ (\alpha, \sigma) $ is : $ \frac {2}{3 ln 2}, 2$. $ f_{\alpha, ...
0
votes
1answer
16 views

Two exponentially distributed random variables w/ different intensity. Which is more probable to take?

Let's say I have two types of light bulbs, A which has $E(A)=100$ hours of lifetime, and B which has $E(B)=200$. I have three of type A and one of type B. I randomly use one of the four, and after 200 ...
0
votes
0answers
11 views

K-wise identical marginal distributions

Suppose I have two joint distributions described by the two sequences of random variables,$X_1, \ldots, X_n$; $Y_1, \ldots, Y_n$. Is there a name/theory/reference for when these two distributions ...
-1
votes
0answers
36 views

The probability density function of the product of independent exponentially distributed RVs

When $X$ ~ exponential distribution(10) and $Y$ ~ exponential distribution(15), and they are independent, find the probability density function of $Z = XY$ I just took an exam for probability ...
1
vote
1answer
16 views

How to find the variance of a normal distribution?

X has normal distribution with the expected value of 70 and variance of σ. It is known that $P(67.36\le X \le 72.64) = 0.34$ find σ So if I understand this right we know that ...
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votes
0answers
24 views

The conditional probability density function with a specific condition [closed]

Assume the following discrete time model: $x(t+1)=Ax(t)+w(t)$ where $w(t)$ is zero mean, iid white noise with bounded covariance matrix $Q$. Let $s=x(t)+x(t-1)+x(t-2)$. How I can find ...
4
votes
0answers
76 views

Suppose $E[X_1] <\infty$. Show that $lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s.

Let $X_1,X_2,X_3,...$ be i.i.d. with $P(X_1 >0)=1$. Define $S_n =\Sigma_{i=1}^{n} X_i$. (a) Suppose $\mathbb{E}[X_1] <\infty$. Show that $\lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s. I ...
2
votes
1answer
33 views

Suppose that the continuous random variables X and Y …

I've tried to attempt all these questions myself first but could someone tell me if these are correct?
2
votes
0answers
29 views

The relationship between random variables, distribution functions and probability measures

Given a probability space $(\Omega,\mathcal{F},P)$, and a random variable $X\colon\Omega\to\Bbb{R}$, we can associate with it its distribution function $F\colon \Bbb{R}\to[0,1]$ defined as ...
0
votes
1answer
19 views

Bivariate distribution $p(x, y) = \frac{1}{n^2}$ for $x = 1, \dots , n$ and $y = 1, \dots , n$. Find the marginal distributions or $X$ and $Y$.

Let two discrete random variables $X$ and $Y$ have joint distribution $$p(x, y) = \frac{1}{n^2}$$ for $x = 1, \dots, n$ and $y = 1,\dots,n$. How would I go about finding the marginal distributions ...
1
vote
1answer
69 views

Y is {-1, 0, 1} and expected value of Y is 0 and the varriance is 1/2 . What is the probability distribution of Y? [closed]

A random variable Y can only take values in {-1, 0, 1}. The expected value of Y is 0 and it's variance is 1/2. Find the probability distribution of Y. I understand the question wants me to find the ...
1
vote
0answers
25 views

Convergence in distribution for two random variables

If $\lim\limits_{n \to \infty} P(X_n\leq T)=P(X\leq T)$ and $\lim\limits_{n \to \infty} P(Y_n\leq T)=P(Y\leq T)$, where $X_1, X_2,\cdots$ and $Y_1, Y_2,\cdots$ are two sequences of random ...
0
votes
0answers
19 views

Simple example of “Spike-and-Slab Prior” for Bayesian Inference

I would really like to understand how Spike-and-Slab Priors work in relation to Linearized Models. Can somebody provide a toy example of a Spike-and-Slab Prior with a Bernoulli spike and a Gaussian ...
1
vote
1answer
22 views

Differential entropy of $\Gamma$

Let $X \sim Gamma(\alpha,\beta)$ be gamma distributed random variable with probability distribution function $$ f_{X}(x)=\frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)},\;x>0 $$ ...
0
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0answers
12 views

Shows Weibull distribution belongs to a one dimensional exponential family

It is given that $f_\eta(y) = h(y)exp(\eta T(y)-A^*(\eta))$ $P_Y(y)= \frac{k}{\lambda} (\frac{y}{\lambda})^{k-1}exp(-(\frac{y}{\lambda})^k)$ What i did was by arranging $P_Y(y)$ to get ...
0
votes
1answer
41 views

Joint pdf of two transformed variables ($W$ and $Z$) from joint pdf of $X$ and $Y$.

Let the joint distribution of $X$ and $Y$ be given by: $f(x,y) = e^{-x}$ if $0 < y \leq x < \infty$ Define $Z = X+Y$ and $W = X-Y$ Find the joint pdf of $Z$ and $W$ Calculate $f_{ZW} ...
0
votes
0answers
24 views

Distribution of a certain stochastic process

Consider on a probability space $(\Omega, \cal F, \mathbb P)$ the following stochastic process on $[0, \infty]$, where $W(t)$ is a Wiener process, all the coefficients $\lambda(t), \mu(t)$ and ...
1
vote
0answers
23 views

Chi-square test to check sampled variance

I have two independent unknown points $x, y \in \mathbb{R}^3$ and a set of $N$ observations $x_i$, $y_i$ of their positions that I model with a normal distribution: $x_i \sim \mathcal{N}(x, ...
2
votes
1answer
34 views

Does the product of three Gaussian random matrices converge in distribution to a Gaussian?

Suppose we have vectors $u,v \in \mathbb{R}^r$ with and matrix $W \in \mathbb{R}^{r \times r}$ where all entries of $u,v,W$ are iid $N(0,1)$. Does the following hold? \begin{equation} \frac{1}{r} ...
0
votes
0answers
34 views

Can I assume a population is discrete if the probability of each event happening is reasonably high?

For instance, if I am presented with data that $P(X=50)= 0.1$, $P(X=60)= 0.2$, $P(X=70)=0.5$, $P(X=80)= 0.2$ Can I assume that it is a discrete distribution, given the fact that if it is a ...
2
votes
0answers
38 views

Let $\Delta_n$ be the smallest distance between any two of these points. Show that $n^{\theta}\Delta_n\rightarrow 0$ in probability.

This is a qual problem。 Let $n$ points be iid uniformly distributed on the unit circle. Let $\Delta_n$ be the smallest distance between any two of these points. Show that ...
1
vote
1answer
18 views

Maximum of poisson process

Let $X^{(1)}_{t\ge 0},...,X^{(n)}_{t\ge 0}$ independent Poisson Processes with common intensity $\lambda$ Find the distribution of the first time that a)at least one event has ocurred in every ...
0
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0answers
21 views

Conditional Probability on Joint Uniform Distribution

This is a very basic question, but I somehow manage to confuse myself all the time. So any help is greatly appreciated. Suppose we have two random variables $X$ and $Y$ with joint distribution ...
0
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0answers
12 views

Scheduling in Wireless Networks with Rayleigh-Fading Interference Simulation Results

I was going through the paper - Scheduling in Wireless Networks with Rayleigh-Fading Interference. Link to PDF: https://people.mpi-inf.mpg.de/~mhoefer/08-0x/Dams14RayleighJ.pdf How to get their ...
1
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0answers
45 views

Integrating a Random Variable and establishing the maximum of a related function

Frequency Regulation of a Power Grid I have a battery that is used to regulate the frequency of a power grid. That is, as the grid frequency varies about it’s ideal value, $f_{nom}$, the battery ...
1
vote
1answer
15 views

Is the difference between the CDFs of two variables the same after addition of third variable to both?

Suppose we have three independent random variables $X$, $Y$, and $Z$ with CDFs $F_X(x)$, $F_Y(y)$ and $F_Z(z)$? Is it the case that $F_{X+Z}(x+z)-F_{Y+Z}(y+z)=F_X(x)-F_Y(y)$? This post provides a ...
1
vote
1answer
19 views

Characteristic function of discret random variable

I try to show the following: Suppose $(X_n),n\geq1$ is a sequence of random variables with uniform distribution on $\{1/n,\dots,n/n \}$. Show that $(X_n)$ converges in distribution to a random ...
0
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0answers
25 views

Density probability

I ask myself a question about of density next : p(xi)=1/(pi*(x²+1)) The law marginal is easy to identify of X and Y: ...
-1
votes
1answer
17 views

Display the quotient between the value of an histograms

i'm triying to display the quotient of two histograms which have the same borders. Is it possible to do is with Matlab ? Thanks a lot !
0
votes
1answer
62 views

Is the given expression convergent as $n\to\infty$?

I want to know whether the following expression is convergent as $n\to\infty$ $$\frac{1}{n}\sum\limits_{k=1}^{\infty}\frac{|\ln n-\ln k|}{k^{(1+1/n)}}\cdot$$ With use of Riemann zeta function ...
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votes
0answers
20 views

Converges a.s. given convergence in distribution

Assume that we have a sequence of R.V. $X_n$ for which the following holds for some $\mu$ $$\sqrt{n}(X_n-\mu)\rightarrow N(0,1) $$ as $n\rightarrow \infty$ where the convergence is in distribution. ...
0
votes
1answer
26 views

What is the probability the lifetime of two electric bulbs with exponential distribution exceeds 2λ?

Let $X$ be the lifetime of a certain electric bulb, and $Y$ the lifetime of its replacement after the failure of the first bulb. Suppose $X$ and $Y$ are independent with common exponential density ...
0
votes
1answer
23 views

Continuous conditional distribution

This is about continuous conditional probability distributions. Why is it that they are allowed to take on single values, while this is a no-no with non conditional continuous distributions(to my ...
0
votes
0answers
45 views

Let $X \sim \text{Exp}(\lambda), Y \sim \text{Exp}(\mu),$ and $Z = X/Y$. Find the pdf of $Z$.

I have random variables $X, Y, Z$, where $X \sim \text{Exp}(\lambda), Y \sim \text{Exp}(\mu),$ and $Z = X/Y$. I calculated the PDF of $Z$ as $$h(z) = \frac{\lambda\mu}{(\mu+\lambda z)^2}$$ Could ...
0
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0answers
24 views

Show that the Z = X/(X+Y) is uniformly distributed on the interval [0, 1], if X and Y are independent exponential r.v. with common parameter λ? [duplicate]

How do I show that the r.v. Z = X/(X+Y) is uniformly distributed on the interval [0, 1], if X and Y are independent exponential random variables with common parameter λ?
0
votes
2answers
46 views

Finding expected number coin flips to get 2 consecutive heads [duplicate]

First, I know what the right answer is, and I know how to solve it. What I'm trying to figure out is why I can't get the following process to work. The probability that we get 2 consecutive heads ...
0
votes
0answers
21 views

What is the probabilty of event A given event B probability in next T time duration?

Assume: a system S with three component: A, B and C. At any moment any component may fail. System MAY fail due to failure of any one component. Given: From the history, we know how many times ...
1
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0answers
17 views

Counterexamples of Cumulative Distribution Function ( multidimensional )

For simplicity, we consider 2-dimensional. We consider the function $F:\mathbb{R}^2\to\mathbb{R}$. Let $F$ satisfies: $0\leq F(x_1,x_2)\leq1$ for $(x_1,x_2)\in\mathbb{R}^2$ ...
2
votes
2answers
63 views

What is the probability distribution of X, if X is the number of times the letter 'e' appears from the set {beware, the, jabberwock, my, son}

A string of letter is chose uniformly at random from the set {beware, the, jabberwock, my, son} Let X be the number of times the letter 'e' appears in the string. Give the probability distribution ...
0
votes
1answer
32 views

Use the delta method to find the distribution of $Z_n$

Let $\overline{X}_n=\overline{X}$ the sample mean such that $\sqrt{n}\overline{X}_n\rightarrow^D N(0,1)$ where $\rightarrow^D$ means converge in distribution. Use the delta method to find the ...
0
votes
1answer
25 views

Simple inequality of tails of random variables

Let $a>0$, $X$ some random variable and $\mathbb{E}[X]=\mu<t$. I was trying to prove the following simple inequality: $$ \textrm{Pr}\left[|X-t|\geq a\right]\leq\textrm{Pr}\left[|X-\mu|\geq ...
0
votes
0answers
6 views

Rank ratio of a signed matrix

Let $A$ be a signed matrix which has entries from set $(-1,0,1)$, and diagonal entries are zero. Consider a matrix $|A|$ corresponding to $A$, which has same entries as $A$ except all $-1$ are ...
0
votes
1answer
19 views

Mean and variance of the sum of cgf gamma and poisson distribution

Suppose that we have the sum of two cumulant generating function: $\log e^{m(e^t-1)} + \log(1-dt)^{-c}$, and we wish to find the expectation and variance without differentiation. I realize ...
4
votes
1answer
32 views

Suppose the pdf of $X$ is $f(x) = \frac{-3}{4} (x-3)(x-5)$. Find the pdf of $X^2/8$.

Suppose that $X$ is a continuous random variable with a pdf $f(x) = \frac{-3}{4} (x-3)(x-5)$ with $3\leq x \leq 5$. What is the pdf of the random variable $Y$, where $Y = X^2/8$. My attempt: ...
0
votes
0answers
23 views

Basic Asymptotic Theory book

I would love if someone can recommend me a book where Basic Asymptotic Theory is thoroughly covered and explained with some examples. I'm currently reading Econometric Analysis of Cross Section and ...