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

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Probability mass function of a degenerate distribution

Wikipedia article "degenerate distribution" states that "The probability mass function does not exist." Is it really right? Why can't it be set as $$ f(x) = \begin{cases} 1 & \quad x = x_0, \\ ...
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7 views

Need help to solve transformation of random variable [on hold]

can anyone help me to solve this problem..it will be really helpful. Thank u
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10 views

Soft question: Distribution of the kth powers of normal random variables.

If $X_1,..,X_n$ are normal random variables then it is knows that: $\underset{i=1}{\overset{n}{\sum}} X_i$ is a normal random vairable and $\underset{i=1}{\overset{n}{\sum}} X_i^2$ is a ...
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2answers
21 views

Bivariate distribution with normal conditions

Define the joint pdf of $(X,Y)$ as: $$f(x,y)\propto \exp(-1/2[Ax^2y^2+x^2+y^2-2Bxy-2Cx-Dy]),$$ where $A,B,C,D$ are constants. Show that the distribution of $X\mid Y=y$ is normal with mean ...
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Origin of the stable distribution name

Mandelbrot (1963) claimed the name stable distribution comes from Paul Levy work: "...The purpose of this paper will be to present and test such a new model of price behavior in speculative markets. ...
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1answer
8 views

Change of variable pdf inverse function

I've been given the following problem: $f(x,y) = e^{-(x+y)}$ on intervals $x \ge 0$ and $y \ge 0$, and $f(x,y) = 0$ otherwise. I'm also given that $Φ_1(x,y) = \frac{x}{y} = U$ and $Φ_2(x,y) = x + y = ...
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1answer
24 views

Bus Problem: Binomial Distribution

For every 10,000 miles driven, the probability that a school bus in the United States will be in at least one accident is 1/6. For 12 buses in the lot, what are the probabilities that: assume that ...
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1answer
27 views

The distribution of the sum of $k$ out of $n$ numbers

Given a list of numbers from $1$ to $n$, I select $k$ values at once (i.e. no duplicates). After summing them up, what is the most frequent value that I am likely to get? My intuition tells me that: ...
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12 views

random number generation from joint distribution where marginal are independents and known

Let $f(x,y)$ be the joint distribution of $X$ and $Y$, where $X$ and $Y$ are both positive and continuous. I can decompose the joint distribution as $f(x,y)=f(x|y) \, f(y)$ and I can easily generate ...
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22 views

concentration of the following random variable: number of items that fit in

This is related to this previous question. Let us assume that we have a capacity $n>0$ which tends to infinity. We are given an i.i.d. sequence of nonnegative random variables ...
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Bounds on functions involving CDFs or Beta function

I have functions of the form \begin{align} I_i = \int_0^\infty F_0(x)^aF_1(x)^b(1-F_0(x))^c(1-F_1(x))^ddF_i(x)~~~~i = 0,1 \end{align} $F_0(x)$ and $F_1(x)$ are CDFs corresponding to the random ...
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1answer
12 views

What is the relation between fix points of random uniform permuation, and probability of independent events occuring.

Let $A_1,\dots,A_n$ be independent events that occur with probability $1/n$ each. Let $p_{n,k}=P($exactly $k$ events occur). One can show with stirlings formula that ...
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1answer
24 views

How can I find the distribution of a stochastic variable X^2 if X is normal standard distributed? [duplicate]

I am considering a stochastic variable X that is standard normal distributed i.e. $$ F_X(x) = \int_{-\infty}^x\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}dt $$ How do I find out the distribution of $X^2$? ...
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23 views

Maximizing P{X=Y} where X and Y are Binomial

X~Binomial(N = 100, p=0.5) Y~Binomial(N = 120, p=0.5) What is the largest possible numerical value of P{X=Y}. X and Y are not necessarily independent.
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2answers
15 views

Does it matter here that random variables are jointly normally 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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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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1answer
26 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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12 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 ...
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2answers
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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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51 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 compute $P_2=P_1 . r$, where ...
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20 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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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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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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25 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
38 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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13 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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12 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 ...
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1answer
13 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, ...
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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 ...
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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 ...
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1answer
43 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 ...
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18 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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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,....
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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 ...
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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
48 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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22 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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21 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 ...
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1answer
43 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 ...
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1answer
33 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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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? ...
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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 ...
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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 ...
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1answer
40 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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27 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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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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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.
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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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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 ...
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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 ...