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

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0
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1answer
9 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 ...
0
votes
1answer
15 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 ...
0
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0answers
8 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 $p(y_i | p) = py_i (1 − p)1−y_i , y_i = 0, 1.$ ...
0
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0answers
9 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
29 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 ...
1
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0answers
25 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 ...
3
votes
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 $X_i >X_{i-1}$, $i>1$ where all realizations follow the same Standard Pareto distribution ...
0
votes
1answer
15 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 ...
0
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0answers
7 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 ...
0
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0answers
11 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
18 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
17 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
9 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 ...
0
votes
1answer
27 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
votes
1answer
38 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?
-1
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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 $x$.
1
vote
1answer
11 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
28 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
18 views

Convolution of uniform random variables [on hold]

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
12 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 ...
0
votes
1answer
24 views

Computing joint probability [on hold]

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
25 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. ...
0
votes
1answer
35 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 ...
0
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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 ...
0
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0answers
31 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
23 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 ...
0
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0answers
12 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 ...
1
vote
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
votes
1answer
13 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 ...
2
votes
1answer
11 views

Joint distribution probabilities

I have a question that is similar to the following(made up here): The construction of a tower of cards is done is two stages, procrastination and the actual building. The time in minutes needed to ...
-3
votes
1answer
35 views

Roll Dice- Expected Winnings [on hold]

I have a problem like this: At a charity game you pay \$1 to roll a die. If you roll a 6, you get \$5. Otherwise, you get nothing. How do I set up a probability distribution and what is the ...
1
vote
2answers
23 views

Normal Distribution finding values

The question says: X is normal with mean -1 and variance 4. Find the value $x_0$ for which the probability is $.2676$ that $X$ will take on a value less than $x_0$. I know this has to deal with ...
0
votes
1answer
22 views

Probability of same last four digits of a telephone number

Hi all this is really my first post here .. Yesterday I was talking to a girl and asked her for her phone number . Once she gave it to me we realized that we got exact same last four digits . Hence ...
0
votes
1answer
32 views

Galton Machine and Unpredictability

We are all familiar with the Galton Machine and the images of the balls cascading through the device and ending up in bins which ultimately show a likeness to the binomial distribution. Most everyone ...
0
votes
1answer
14 views

Probability of 2 of three independent events occuring

Three objects are thrown at a target. The probabilities the individual objects will connect with the target is .75, .85 and .90. Find the probability that at LEAST two of the objects hit the target? ...
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0answers
13 views

Waiting time probability question

I want to solve the following problem: A dentist works 4 hours a day. Patients arrive on the average of 1 per 20 minutes and one patient spends on average 15 minutes with the dentist. Both time ...
0
votes
1answer
27 views

proof of property of exponential distribution, using taylor polynomial

I want to prove that if we have an exponential distribution with parameter $\lambda$, we have that $P(X \le x)=\lambda x + o(x)$. I want to do this by using Taylor-series and the lagrange remainder ...
0
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0answers
20 views

p.d.f and distribution of multivariate normally distributed variables

Suppose $X\sim N(\mu,V)$ where $\mu = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix}$ $V = \begin{pmatrix} 3 & 2 & 1 \\ 2& 4 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ a) ...
1
vote
2answers
23 views

Find Normalizing constant

let $f(x,\theta)=C_\theta \exp(-\sqrt{x}/\theta)$ where $x$ and $\theta$ are both positive. Find the normalising constant $C_\theta$. I get $C_\theta=\sqrt{2}/\theta$ but my book says ...
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0answers
32 views

Average minimum distance

I posted a question earlier here and someone pointed out that it might not be possible to find a closed form solution due to the elements of $\mathbf{g}$ and $\mathbf{f}$ defined below coming from a ...
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0answers
7 views

Singular distributions: Applications and Instances

This is the duplication of the question I asked here. I repeat it here with hope of getting new answers. Singular distributions are special mathematical objects. They have an interesting property ...
0
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1answer
72 views

Challenging question about probability [closed]

A manufacturer of plastics claims that its waste is managed in such a way that benzene, a harmful chemical, cannot get into the local ground water. People living near the factory are not so sure. A ...
0
votes
3answers
61 views

Finding probability of a random point [closed]

Consider a square with sides of length 1 and the bottom left corner at (0;0). Choose a point P randomly within the square. Show that the probability that P is closer to (0;0) than to (0.5, 0.5) is ...
0
votes
1answer
14 views

Distribution of a function of a uniform random variable.

I ran across this example the other day and was surprised at how stumped I was. Suppose $U$ is a uniform random variable on the interval $[0,1]$. Let $F = \frac{1}{U+3}$. What is: ...
0
votes
1answer
20 views

Joint distribution and Integration

I was trying to prove a problem in my notes and now I need to whether prove or disprove the following claim: Assume $X,Y,W,Z$ are random variables defined on $(\Omega,\mathcal{F},P)$. If $(X,Y)$ and ...
0
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0answers
29 views

Average minimum distance between two random vectors

Let $\mathbf{y_1} =\begin{bmatrix}g_1x_1 & g_2x_1 & \dots & g_Nx_1 \end{bmatrix}$ and $\mathbf{y_2} = \begin{bmatrix} f_1x_2 & f_2x_2 & \dots & f_Nx_2\end{bmatrix}$. All the ...
0
votes
0answers
20 views

Queueing and probabilities

Messages are transmitted from low speed terminals and arrive at a message concentrator at a Poisson rate of $600$/hour. They are held in a buffer until a hi-speed trunk line is free to transmit them. ...