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

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5 views

Derivation of Chi-Square Distribution using Dirac Delta Distribution

I'm trying to understand a derivation of the density of a Chi-Square distribution with one degree of freedom that my professor gave in class. Here are the steps: \begin{align} p(y) &= ...
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1answer
19 views

Biostatistics math

It has been reported that the average monthly cell phone is $\$50.00$. Assuming a normal distribution and a standard deviation of $\$10.00$, what is the probability that randomly selected cell ...
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1answer
21 views

Probability math problem. [on hold]

Certain data obtained from a study of a group of 1000 subscribers to a certain magazine relating to their sex, marital status, and education were reported as follows: 312 males, 470 married, 525 ...
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1answer
25 views

Continuous Probability Density Function

The problem is: Verify that the function p: [0,2] [0,1] given by p(X) = {X, if 0 < X < 1 2 - X, if 1 < X< 2 is a ...
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1answer
12 views

Hypothesis testing on minimum of exponentially distributed random variables

I am completely stuck with the following problem, because I do not know how to start: Let $X_1,...,X_n$ be independent and exponentially distributed with unknown parameter , and let ...
0
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1answer
20 views

Sum of random variables and joint distribution

Consider the following contingence table: $X_1, X_2, X_3 and X_4$ are r.v with independent Poisson distribution with parameters $(\lambda_i)_{i=1,...,4}$. Show that, (a) if $N\geq 0$ is a given ...
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0answers
24 views

Help with finding $E((\Phi(aX+b)^2))$ when $X$ is standard normal

I need help with calculating $$ E((\Phi(aX+b))^2)=\int_{-\infty}^\infty (\Phi(ax+b))^2\phi(x)dx $$ where $X$ is a standard normal random variable and $a$ and $b$ are constants. $\Phi$ and $\phi$ ...
2
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1answer
26 views

Source needed: Does asymptotic normality yield asymptotic unbiasedness and consistency?

Assume that $$\sqrt{n}(\hat g - g(\theta)) \xrightarrow{d} Z, $$ where $Z$ is $N(0,\sigma^2)$. Does this already imply asymptotic unbiasedness and/or consistency, i.e., $$ E[\hat g] \rightarrow ...
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1answer
20 views

If I have a random variable X with a given Probability Density Function, How do I find the PDF of the area of a circle with radius X?

To find the PDF of the area of the circle, do I just substitute the PDF of the random variable X in for the radius in the circle area equation?
3
votes
2answers
29 views

What's the densitiy of the product of two independent Gaussian random variables?

Suppose that $X,Y$ are two scalar independent normal random variables, $X \sim N(\mu_X,\sigma_X^2)$, $Y \sim N(\mu_Y,\sigma_Y^2)$. I'm particularly interested about the case where we don't assume ...
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0answers
12 views

combining continuous distributions

If we have several continuous distributions, for example ten Beta distributions, how we can combine them by the linear and log-linear opinion pool methods? I know how to combine discrete ...
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1answer
15 views

Showing the distribution of a poisson process

A large lump of radioactive material has a long half life. Let $D(t)$ be the total number of decays which occur in the radioactive material in the period of $t$ hours starting at noon on a particular ...
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2answers
49 views

How to characterize rotations in $\mathbb{R}^n$?

I am studying the performance of an optimizer algorithm to find the $$ \textrm{argmin}_{x\in \mathbb{R}^n} f(x) \text{ where } f : \mathbb{R}^n \rightarrow \mathbb{R} $$ I would like to test how the ...
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0answers
15 views

Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go: Let $\theta _{i},$ $i=1,...,N$, be drawn independently ...
0
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1answer
32 views

How to find E(XY) and g(x)&h(y) from f(x,y)? [on hold]

When $0\le y \le x \le 1$ and $f(x,y) = 8xy$, how can I find $E(XY)$ and marginal functions $g(x)$ and $h(y)$?
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1answer
12 views

T-Intervals and % Confidence Interval

The question is the following: " A random sample of six 2009 sports cars is taken and their "in the city" miles per gallon is recorded. The results are as follows: 23 19 24 17 16 22. Assuming the ...
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votes
3answers
29 views

Integrating the product of Poisson and exponential pdf

So I'll spare the background as to why, but I'm trying to integrate the following: $$\int_0^{\infty} \frac{e^{-(\lambda+\mu)t}(\lambda t)^n}{n!} dt$$ If you parameterize a Poisson w/ $\lambda$ and ...
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1answer
19 views

Lack of memory property of probability distributions

According to wikipedia lack of memory property applies to geometric and exponential distributions. I was trying to apply it to binomial distribution. Am I modelling my question correctly? So imagine ...
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1answer
12 views

Derive the Cramer-Rao lower bound (CRLB) for any unbiased estimator of $\mu^2$.

Let $Y_1, Y_2, . . . , Y_n$be a random sample from a normal distribution with mean μ and variance 1. Derive the Cramer-Rao lower bound (CRLB) for any unbiased estimator of $\mu^2$. Could anyone ...
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votes
1answer
20 views

Sum of two independent non-identical uniform random variables

Let's say we have two independent random variables, $X$ is uniform on $[0,1/2]$ and $Y$ is uniform on $[1/2,1]$. If we look at the distribution of $X+Y$, is it triangular distribution between ...
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0answers
12 views

Questions related to Rao–Blackwell theorem

In this exercise, we illustrate the direct use of the Rao–Blackwell theorem. Let $Y_1, Y_2, . . . , Y_n$ be independent Bernoulli random variables with $p(y_i | p) = py_i (1 − p)1−y_i , y_i = 0, 1.$ ...
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0answers
27 views

find E($\bar{Y^4})$ by using moment generating function for a normal distribution with mean μ and variance 1.

Let $Y_1, Y_2, . . . , Y_n$be a random sample from a normal distribution with mean μ and variance 1. I would like to find E($\bar{Y^4})$ by using moment generating function. The setup I have right ...
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0answers
31 views

A Simple yet interesting “function of a random variable” question

Given continous density functions $f_0,f_1$ on $\mathbb{R}$ and $Y$, a random variable following the density $f_0$, I am able to calculate the density function $h$, of $\ln l(Y)=\ln(f_1/f_0(Y))$ as ...
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0answers
42 views

Need help finding probability distribution [on hold]

In Cairo $30\%$ of residents listen to the local fm radio. $10$ residents are chosen at random: a) state the distribution of the random variable b) find the smallest value of $s$ so that $\Pr(X \ge ...
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1answer
1k views

Distribution of the sum of the $q$th largest observations to the sum of total for a power-law.

Where $X_{(1)}, X_{(2)}, \ldots,X_{(n)}$ are sorted independents r.v.s, where we index and order in such a way that $X_{(i)} \geq X_{(i-1)}$, $i>1$ where all realizations follow the same Standard ...
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vote
2answers
24 views

Why normal approximation to binomial distribution uses np> 5 as a condition

I was reading about normal approximation to binomial distribution and I dunno how it works for cases when you say for example p is equal to 0.3 where p is probability of success. On most websites it ...
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0answers
10 views

How to properly clamp Beckmann Distribution

I am trying to implement the Cook-Torrance Microfacet BRDF shading model and I am having some trouble with the Beckmann Distribution: Beckmann Distribution with width parameter ...
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0answers
15 views

Beta marginal distributions and Dirichlet distribution

I know Dirichlet distribution has Beta marginal distributions (sum to 1). I am not sure if the other direction is also correct. That is, for example, if we have 3 beta distributions: $X_1 $ ...
0
votes
1answer
22 views

Pivotal quantity of Weibull distribution

If I have $X_{1},\ldots,X_{n}$ a random sample from a Weibull distribution $X\sim WEI(\theta,2)$.How can I show that $Q=2\sum\limits_{i=1}^n X_{i}^2/\theta^2\sim \chi^2(2n)$. I have not learnt any ...
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0answers
18 views

Normal distribution exercise [on hold]

In a factory, compacts are filled with a cosmetic powder. We consider the weight of the powder follows a normal distribution $N\sim(\mu, 1.21)$. The value of $\mu$ depends on the setting of the ...
2
votes
1answer
10 views

Bound on expectation of function of standard normal, $\mathbb{E}[\exp(Z^a)]$

I'm trying to find the maximum (or sup) of the value of $a$ such that $$\mathbb{E}[\exp(Z^a)]<+\infty$$ where $Z\sim \mathcal{N}(0,1)$. Obviously for $a=1$ the expectation is finite since it is the ...
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votes
1answer
29 views

Average of IID Cauchy RVs

Suppose that $X_i$'s are iid Cauchy RV's with pdf $f_u (x) = \frac{1}{\pi} \frac{u}{u^2+x^2}$. I am aware that the RV $Y:=\frac{1}{N}\sum_{k=1}^N X_k$ has the same density as the $X_i$'s. I am trying ...
2
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1answer
39 views

what is the distributions of the random variable?

If moment generating function is $m(t)=[(1/3)e^{t}+(2/3)]^{5}$, then what is the distributions of the random variable?
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1answer
29 views

A basic question on uniform distribution [on hold]

I want to know under what condition on random variable $X$, $\{\log_{10}X\}$ is uniformly distributed. Here $\{x\}$ is the fractional part of $x$.
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vote
1answer
13 views

Poisson process - number of store purchases in a given time

Customers enter a store according to a Poisson Process of rate = 6 per hour. Individuals who enter the shop have (independently of each other) probability $\theta$ of buying something. ...
1
vote
1answer
32 views

Conditional expectation and Rao Blackwell

Consider a family of densitites $f(x,\theta)=\frac{\exp(-\sqrt{x})}{\theta}$. Let $X_1$ be a single observation from this family. I have shown that $\sqrt{X_1}/2$ is an unbiased estimator. Now ...
0
votes
1answer
19 views

Convolution of uniform random variables [closed]

Let $X$ and $Y$ be IID $U[0,1]$ random variables. Find $\text{Prob}(0 \leq X^2 < Y < X^{0.5} \leq 1)$.
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0answers
16 views

Distance of random span to a vector

I've been batteling with the following problem: Assume we have a diagonal matrix $D \in \mathbb{R}^{l \times l}$, a vector $\beta \in \mathbb{R}^l$. Next we simulate a random matrix (Idea inspired by ...
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1answer
25 views

Computing joint probability [closed]

Let $X,Y\sim \text{Exp}(1)$ (exponential random variables with parameter $1$). Then prove that $$Pr(X> z_1, \frac{Y}{X} > z_2) = \dfrac{e^{-z_1 (1+z_2)}}{1+z_2}, \forall z_1,z_2>0$$
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2answers
29 views

If $X$ is a continuous random variable uniformly distributed over $[a,b]$, then is $Y=2-4X$ uniformly distributed over $[c,d]$? Why?

I ran into this problem solving one of the problems on my course and if I knew that this applies and how to simply prove it, it would help me a great lot.
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0answers
24 views

Probabilistic model of parallel web servers

Note: The following probabilistic model of parallel web servers is abstracted from an engineering project. I am not good at probability theory and I am seeking some evaluations and suggestions. ...
1
vote
1answer
37 views

Hammersley–Clifford theorem

I'm reading this paper http://image.diku.dk/igel/paper/AItRBM-proof.pdf and I got stuck in page 4 with equation (1) that's based on Hammersley–Clifford theorem. I'm not good in reading set theory ...
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0answers
29 views

$E[X]< (\sum_{n=0}^\infty P[X>n]< E[X]+1$

If X takes only non-negative integer values then I figured out $$E[X]= (\sum_{n=0}^\infty P[X>n]$$ but I'm having hard time proving $$ E[X]⩽ (\sum_{n=0}^\infty P[X>n] ⩽ E[X]+1$$ for any ...
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0answers
32 views

Show that $Y_i$ is independent of $Y_j$ for any $i$ not equal to $j$

Let $\{X_1,X_2,\ldots\}$ be independent, identically distributed, absolutely continuous random variables. Let $Y_n=I\{X_n>\max(1< i < n)\}$ for $n=2,3,\ldots$ a) Show that $Y_i$ is ...
3
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0answers
24 views

Write $\Phi_n(\sqrt{y-1})$ in terms of $\Phi(y)$ and $n$. ($\Phi_n$ CDF of a $\mathcal{N}(0,\frac{1}{n})$)

I'm trying to solve the following problem: Let $X_n \sim \mathcal{N}(0,\frac{1}{n})$, and let $Y_n$ be the variable defined by: $$Y_n(\omega)=\int_{-1}^1 | X_n(\omega)-t |\,dt $$ Let $F_{Y_n}$ ...
0
votes
1answer
21 views

Formula needed for calculating probability of recurring events

I'd like to find an answer for calculating the following recurring events: You have X opportunities of picking a ball from a sack. Every time after a ball is picked, the ball is returned to the sack. ...
0
votes
1answer
25 views

Average number of rolls before going broke

I have a difficult probability question to resolve. Say you have 2 chances to roll a dice. If you roll a 6, you're awarded 2 additional rolls. You can receive infinite number of additional 2 rolls ...
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0answers
13 views

Bayesian mean square error

Given a i.i.d sample $X_{1},..,X_{n}$ of bernoulli random variables test 2 hypotheses $H_{0}:p=2/3$ and $H_{1}:p=1/3$. Bayesian prior is $\pi(2/3)=1/3$ and $\pi(1/3)=2/3$. Find the bayesian criterion ...
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1answer
24 views

$X$ and $Y$ have a joint distribution density function. Working out a marginal density function for $X$ and $Y$

$f_{X,Y}(x,y) = \frac{3}{2}(x^2+y^2)$ if $0 \lt x \lt 1$ and $0 \lt y \lt 1,$ or $0$ otherwise. I want to find the marginal probability density function of $X$ and $Y$ and then find $Pr(0 \lt x \lt ...
2
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1answer
15 views

Expected value, variance and probability from a joint distribution function

Lets say I am given the following table that shows the joint probability function of X and Y: $$\begin{array} \\{}&y=1&y=2&y=3 \\x_=1&0.1&0.2&0.1 ...