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

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2answers
9 views

Does it matter here that random variables are jointly distributed?

My lecture notes ask the following (true/false) question on understanding: Jointly normally distributed random variables are independent iff they are uncorrelated. I don't quite understand what ...
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0answers
11 views

Joint density function of exponential and gamma distribution

My problem is: $X_1,...,X_n$ are independent exponentially distributed random variables with $\lambda=1$ paremeters. I have to find the joint density funcitions of $ Y=\sum\limits_{i=1}^n{X_i}$ ...
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0answers
9 views

What is the variance of this random variable: number of items

Let us assume that we have a capacity $n$ which tends to infinity. We have an infinite number of random variables $X_1, X_2, \dotsc$, where each $X_i$ is independent and identically distributed with ...
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0answers
10 views

Verification of convolution between gaussian and uniform distributions

Let $n \sim \mathcal{N}(\mu, \sigma^2)$ and let $u \sim \mathcal{U}(a,b)$, with $b>a>0$, and suppose that $n$ and $u$ are independent random variables. Let $g = n + u$. The probability density ...
1
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2answers
13 views

Why does marginalization of a joint probability distribution use sums?

I'm going through a book that talks about probability distributions. The part which is tripping me up is conceptual. It says: "We can recover the probability distribution of any single variable from ...
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0answers
23 views

To make a polynomial with coefficients in a finite field uniform at random

We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$. Let $P_1$ be a polynomial such that $P_1 \in R[x]$. The aim is to make $P_1$ uniformly at random. ...
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0answers
13 views

How to calculate Fourier Transform of logarithmic function?

Given a random variable (RV) $S$ equal to the sum of two mutually independent (RVs) $X_1,X_2$,i.e.$S=X_1+X_2$ and piece-wise probability density functions (PDFs) of $f_{X_1},f_{X_2}$ are as follow: ...
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0answers
8 views

variance of binomial distribution how to work with explaination [on hold]

an experiment involves rolling a fair die 10368 times what would the variance of the number of times a number less than 3 is rolled? I have no clue how to even start to work this problem. I am ...
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0answers
9 views

Relation between Bernoulli RV, binomial RV, geometric RV and Poisson RV [on hold]

what is the relation between Bernoulli RV, binomial RV, geometric RV and Poisson RV? And how we represent them?
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0answers
22 views

Trouble with Conditional probability and expectation [on hold]

I have a few questions in probability that have been bothering me. The first is this: Why is $$E(T-t | T \ge t) = \int_t^\infty \frac{(s-t)f(s)~ds}{P(T\ge t)}. $$ The second is this: How does one ...
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2answers
24 views

Expected Number of points in Point Poisson Process

Let $\lambda$ be the intensity of points, distributed as point poisson process, in a circle of area $R$. Then, the Cumulative disributive function is given by: $$ P(r \leq R) = 1 - e^{-\lambda \pi ...
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0answers
10 views

How do I describe probabilities of realised variables in terms of CDFs and PDFs?

I have a game theory problem here where the realisation of the random variable $Y₂$ (termed $y₂$ is observed in the first stage of the game. $Y₂$ is the second highest order statistic for $n-1$ ...
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0answers
11 views

Help in finding the functional form of the probability density function

This may seem trivial but I will appreciate help in determining the functional form of the probability density function (pdf) for the following case. Will highly appreciate some guidelines on how to ...
2
votes
1answer
12 views

Beta/Dirichlet question

A generalization of the beta distribution is the Dirichlet distribution. In its bi-variate version, (X,Y) have pdf $f(x,y) = Cx^{a-1}y^{b-1}(1-x-y)^{c-1}, 0<x<1, 0<y<1, ...
1
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0answers
10 views

maximum likelihood of a dirichlet prior

Suppose $\theta \sim D(\alpha)$ where $D$ denotes the Dirichlet distribution and $\alpha = (\alpha_1,\ldots,\alpha_K)$ its hyperparameter, in which case: $$p(\theta) = \frac{\Gamma(\sum_k ...
1
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2answers
32 views

Probability of Snow in New York

In New York, snow is reported 25% of days in February. If this trend continues, what is the probability that it will snow exactly 9 days this coming February and is not a leap year? Solve this ...
1
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1answer
29 views

Show independence of number of heads and tails

I am independently studying Larry Wasserman's "All of Statistics" Chapter 2 exercise 11 is this: Let $N \sim \mathrm{Poisson}(\lambda)$ and suppose we toss a coin $N$ times. Let $X$ and $Y$ be the ...
0
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0answers
14 views

Expected number of points within a defined radius [duplicate]

I have the following probability distribution function $$ P(r \leq R) = 1-e^{\lambda \pi R^2} $$ where $lambda$ is the intensity of a point poisson process. I would like to calculate the $\lambda(R)$. ...
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0answers
15 views

Nondecreasing sequence of a set [on hold]

Find $\lim_{x\to\infty}C_k =$ {$x:\frac1k \le x \le3-\frac1k$},$C_k =$ {$(x,y):\frac1k \le x^2+y^2 \le4-\frac1k$} where k= 1,2,3,....
0
votes
1answer
24 views

Letter Arrangements of M,A,R,Y

List all possible arrangements of the four letters m,a,r,and y. Let $\; C_1 \;$be the collection of the arrangements in which y is in the last position. Let $\; C_2\;$ be the collection of the ...
1
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1answer
38 views

Uniformly at random polynomial

We have a polynomial of degree $d$, and multiply it by a polynomial whose coefficients are chosen uniformly at random and its degree is equal to or less than $d$. My question is whether the result is ...
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1answer
47 views

Birthday Problem: Big Numbers and Distribution of the Number of Samples involved in Collisions

A lot of questions about the birthday problem can be found here, but none seems to address my problem: Background I am thinking of a hash-type data structure design which accepts a certain number of ...
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0answers
18 views

cantor staircase function uniform distribution on cantor set

suppose Cantor staircase function $F$ is extended to have $F(a)=0$ for $a<0$ and $F(a)=1$ for $a>1$. Then how can one show that $F$ is the cumulative distribution function of the uniform ...
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0answers
19 views

Finding the roots of a polinomial function obtained by a Binomial c.d.f.

I came across with the following question and I am also attempting to solve it. Let $B(K/2;K,1-x)$ be the Binomial c.d.f. with $K$ trials having at least $K/2$ success with each trial having success ...
2
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1answer
41 views

How to calculate conditional probability with inequality

I know that: \begin{equation}\displaystyle P(A=x|A+B=y) = \frac{P(A=x \cap A+B=y)}{P(A+B=y)}\end{equation} Assuming $A$ and $B$ are independent, the intersection of the two events can be resolved as ...
0
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1answer
31 views

Z~U[0,1] and X=f(Z) and f is:

I have found the f(z): Now, I need to find pdf of X. And I can see that 0< f(Z)=X<1, I don't know how I am going to get f(X), I just can see that f(X)=0 when X<0 and x>1, but I can see a ...
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0answers
4 views

Covariances of variables in a linear gaussian bayesian network (more than one new variable)

My question is closely related to the following question: Linear transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable? ...
0
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2answers
17 views

Simple finding the PDF given function

I am a little confused on how to go about finding the PDF given a condition for a function. So I have the function $$ Y(x)=ae^{-bx} \,\,\,\,\,\,\, a,b,x \geq0 $$ and I need to find the value for X ...
0
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0answers
12 views

approximation using a shifted gamma

I have a PDF that is weighted sum of a gamma and shifted gamma distribution f(x)=0.75*gamma(x,100,0.1)+0.25*gamma(x-10,1,10) is it possible to approximate this PDF by a shifted gamma that has the ...
1
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1answer
23 views

Central limit theorem kind of statement for records

I am trying to prove the following statement, but I do not know how to go on: Let $F(x)$ be an arbitrary continuous distribution function. Then there are constants $A_n, B_n > 0$ such that, as ...
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0answers
21 views

Discrete Random Variable and Its Probability Distribution

EZ Language Center offers a 2-month summer course on three of the most popular and romantic languages aroun the world. French, spanish, and italian. Their database shows that .27, .40 and .33 of their ...
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0answers
27 views

Normal probability [on hold]

Female baby moss grow faster than male baby moss. In this question you will investigate the distribution of carapace (shell) lengths for baby moss of one particular species. In their early weeks, ...
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0answers
9 views

mean and autocorrelation of random process [on hold]

X(t)=Acos(2πft+Θ) where Θ and f are constant. A: uniformly distributed over the interval (α,β) Find the mean and autocorrelation of X.
2
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0answers
34 views

Relationship between chi-squared and standard normal distributions.

It is well known that if $Z \sim N(0,1)$ then $Z^{2} \sim \chi^{2}(1)$. However, if we know that $X^{2} \sim \chi^{2}(1)$, under what conditions is it true that $X \sim N(0,1)$? As far as I know, this ...
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0answers
9 views

A cell dies at a constant rate r and what is the density function of its life time.

The problem: A cell dies at a constant rate $r$ and the its life time is the duration from t=0 to when when it dies. what is the density function of its life time $l$? I have done some relevant ...
0
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1answer
13 views

Using a joint distribution table to find probability?

I have the following joint distribution table. I am trying to answer the following questions. A,B,C,D For (a) I put $P(X=1, Y=2)=1/20$ (B) $p(x=0,1\le y<3)= 1/4+1/8$ But I am not sure how to ...
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2answers
23 views

Does the sampling distribution coincide with the population distribution if every possible sample is taken?

Say you have a population. You take random samples repeatedly, and the distribution of all the means of those random samples is the sampling distribution. Right? So does that mean, that if you take ...
3
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0answers
34 views

Distribution of $\frac{X}{|Y|}$, where X and Y are standard normal r.v.'s

Let X and Y be independent standard normal random variables. What is the distribution of $\large \frac{X}{|Y|}$? Attempt: Let $\large U = \frac{X}{|Y|}$ and $ V = |Y|$. This transformation is not ...
0
votes
1answer
17 views

Finding the cumulative distribution [duplicate]

How can I find the cumulative distribution function for the following prob density. $$ f(x) = \begin{cases} x & \text{if } 0<x<1 \\ 2-x & \text{if } 1 \leq x <2 \\ 0 & ...
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1answer
23 views

Probability question using Poisson

Here is my Question: A country bus driver picks up passengers randomly and independently at a mean rate of 12 per hour. (i)Find, correct to 3 decimal places, the probability that he picks up ...
1
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1answer
35 views

Joint density calculation-Spot the error

Suppose $X_1 $ and $X_2$ are i.i.d standard normal r.v.s and $Y=X_1^2+X_2^2$, then we know $Y \sim \chi_2^2 $ and $f_Y(y)= \frac{1}{2}e^{\frac{-y}{2}}$. Using the identity $f_{X,Y}=f_{X\mid Y} \cdot ...
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0answers
5 views

Asymptotic distribution of MLE of the parameters for an ARMA(1,1) model

I was just have trouble interpreting and understanding the asymptotic distribution of the MLE of the parameters for and ARMA(1,1) model (and an ARMA(p,q) model in general). It has been given that ...
0
votes
1answer
29 views

Binomial Distribution Proof

What is that $I(\cdot)$ in the 3rd step means? $p_{x_n}(y_n-y_{n-1}) = p(X_n=y_n-y_{n-1}) = p(X_n)$ belongs to the interval $\{0,1\}$, since it is a random variable. Then, how are getting to that ...
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0answers
18 views

Upper and lower bound for variance given mean and median

I have a random variable $X$ taking values in the interval $[0,1]$ with mean 0.2 and median 0.3. What are the lower and upper bounds of the set of possible variances of $X$? I am able to solve this ...
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votes
1answer
26 views

Bernoulli distribution solving for n

So we have this missile protection system that has $n$ radar sets that are all independent. Each have a probability of $0.9$ of detecting a missile. How large must $n$ be if we want the probability ...
0
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2answers
24 views

Probability to pass multiple-choice test, with two type of questions

First i want to say there are a lot of questions related to this, but i couldn't find a similar case. Suppose we have the typical problem where we need to compute the probability of pass a ...
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0answers
15 views

Sum of the smallest or greatest k components of a random vector drawn from a symmetric Dirichlet distribution? [on hold]

Is any distribution known for the sum of the smallest or greatest $k$ components of a random vector drawn from a symmetric Dirichlet distribution?
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0answers
38 views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
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2answers
45 views

Uniform Random Number

Two uniform random numbers are chosen one after the other. what is the probability of second number second random number greater than first number? I tried this way Please correct me if I am wrong. ...
1
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2answers
21 views

Integrating a function wrt different measures [duplicate]

Suppose that $(\Omega, \mathcal E, P)$ is a probability space and $X\colon \Omega \to \mathbb R$ is a RV defined on $\Omega$. Denote as $\mu\colon \mathcal B \to [0,1]$ the probability measure on ...