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

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1answer
86 views

Probability ( kth order statistics <x , (k+1)th order stat >x)

consider $n$ iid draws according to some cdf $F$. What is the probability of the following event: the $k^{\text{th}}$ highest value is smaller than $x$ AND the $k+1$ highest value is larger than $x$. ...
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1answer
41 views

Most probable probability distribution given outcome

Given several values of x that are outcomes of a random experiment and a parametric model of the probability, what is a sensible way to choose the parameters? Does it make sense to optimize the ...
0
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1answer
42 views

$X_i$ are Gaussian random variables with mean $0$ and variance $\sigma$

Let $X_1,...,X_k$, $k\geq 2$, be independent random variables, each having the same positive and differentiable density function $f$. Further suppose $\prod_{j=1}^k f(x_j)$ depends only on ...
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0answers
55 views

Intuition behind Stationarity in Delayed Renewal Processes

I was going through excess life and renewal processes in my notes when I came across a proposition in my notes that said that given a delayed renewal process X with independant interarrival times ...
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2answers
40 views

What is the probability of uniformly sampling a point in d-dimensional hypercube?

Let us consider a hyper-cube whose length is l units along each of its d-dimensional structure. It is desired to uniformly sample a point inside the hyper-cube. How to do uniform sampling and what ...
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1answer
20 views

Bivariate Poisson Distribution - how can a specific case be computed?

I wanted to generate a bivariate Poisson distribution (D1,D2) and to do so, I did according to the indication found on wikipedia (bottom of the page, ...
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0answers
219 views

Compound normal distribution with mean from truncated normal

I'm trying to compute the distribution of a compound normal where the mean is drawn from a truncated normal. To be specific, I want to find the unconditional distribution $p(\theta)$, when the ...
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1answer
18 views

Conditional probability with four random variables

Assume A, B, C, and D are i.i.d random variables and k is a fixed constant. I want to find $\textbf{P}(A < B, C, D | D = k)$. How would I go about getting this, in terms of the cdf of these random ...
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1answer
107 views

Suppose that number of mistakes on a page is a Poisson RV and independent. From $n$ pages, find the expected number with no mistakes?

A textbook has $n$ pages. The number of mistakes on each page is a Poisson RV with parameter $\lambda$ and is independent of the number of mistakes on all other pages. What is the expected number of ...
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1answer
163 views

Expected value of a function of a random variable [duplicate]

Let X be a random variable whose PDF is $f(x)$, and $g$ a function of random variable X. I want to prove that $$E[g(X)] = \int{g(x)f(x)dx} $$ I've perfectly understood it in discrete case and I ...
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0answers
40 views

Order statistics: does distribution of sum of two of them uniquely determine parent distribution?

Let $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d. r.v. with bounded range (say, the interval [0,1]), with cdf $F$. Let $Y_1 \geq Y_2, \ldots, \geq Y_n$ be the corresponding order statistics. My ...
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1answer
22 views

How to know the result of entropy function using uniform distribution set

In the entropy function here $H(s) = -\sum P(class=i|S)log_2{P(class=i|S)}$ I am trying to understand what is the domain of it's output for any input. I know that given a set where the frequency of ...
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1answer
70 views

Intro to probability chapter 4 ex 31

A group of 50 people are comparing their birthdays (as usual, assume their birthdays are independent, are not February 29, etc.). Find the expected number of pairs of people with the same birthday, ...
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1answer
132 views

A proof which results in Gamma (or Erlang) distribution-From Karlin & Taylor's “A First Course in Stochastic Processes”

The random variables X and Y have the following properties: X is positive, i.e., $P\{X > 0\} = 1$, with continuous density function $f_X(x)$, and $Y\mid X$ has a uniform distribution on $\{0,X\}$. ...
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0answers
22 views

Total probability distribution of multiple random lotteries

My question: Imagine $d$ identical lotteries. Each individual lottery picks a cost $c_{i}$ between $0$ and $1$. Picking a costs occurs with probability distribution $f(c)$. The total cost of these ...
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1answer
23 views

Given the distribution of $X$, whats the distribution of $cX$

Let's say $X \sim \chi_k^2(\lambda)$ with pdf $f_x(x)$ (i.e. noncentral chi-squared distribution). What can we say about the distribution of $Y = cX$ ? where $ c \in \mathbb{R}^+$ I know that $f_y(y) ...
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2answers
29 views

What is the best choice for $\mu$

A bottle is underfilled if it is less than $500$ml and overfilled if it is greater than $560$ml. Suppose the purchaser fines the supplier ${$}1$ per underfilled bottle. It costs the manufacturer ...
3
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0answers
97 views

Sufficient statistics and UMVUE for joint poisson, bernoulli

Given a pair $(X,Y)$ of r.v.s such that: $$X \sim \text{Poisson}(\lambda)\quad \text{and}\quad Y \sim B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a one-dimensional ...
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3answers
62 views

How to prove $\sum_{i=1}^{n}\binom{n}{i}p^i(1-p)^{n-i}i = np$?

How to prove, when $p\in[0, 1]$, $$\sum_{i=1}^{n}\binom{n}{i}p^i(1-p)^{n-i}i = np$$ Is there a name for this formula?
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2answers
44 views

Find the mean, variance and $f(21)$ of this if it is a p.g.f.

$$G(t) = t^{11}e^{9t-9}$$ Find the mean, variance and $f(21)$ Please help me with find $f(21)$ as easily as possible. For the mean and variance: $$G'(t) = 11t^{10}e^{9t-9}+9t^{11}e^{9t-9}$$ ...
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3answers
214 views

If the joint distribution is uniform, then the random variables are independent?

This is a problem that I am stuck at. If $X_1$ and $X_2$ are independent, it would be easier. But, the problem asks me the converse. For (i), I suspect that $X_1$ and $X_2$ are independent. But I ...
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2answers
79 views

Evolution of a discrete distribution of probability

I am designing a virtual card game and I defined an evolution of probabilities, but I don't have the knowledge on this matter to find out how they will evolve. I hope you help me here, with ...
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2answers
44 views

Expected Value and Variance of transformed Random variable

I am trying to find the expected value and variance of $Y_i=\ln(X_i)$ for $X$ is uniformly distributed between $1$ and $3$. I believe that $E(Y_i)=(\ln3)/2$ and $\operatorname{Var}(x)=(\ln3)^2/12$. ...
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0answers
29 views

How to choose assymetry for KL divergence?

I have two 2D probability distributions of eye movements of two different images. Suppose I call the first distribution of Image 1: $P$, and the second distribution of image 2: $Q$. Since ...
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2answers
91 views

Suppose $X$ and $Y$ are independent Poisson random variables. Find the conditional probability mass function $P(X=k\mid X+Y=m)$ [duplicate]

Suppose $X$ and $Y$ are independent Poisson random variables with parameters $\lambda$ and $\mu$, respectively. Find the conditional probability mass function $P(X=k\mid X+Y=n)$. Don't know how to ...
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0answers
44 views

Joint characteristic function of $X$ and $F(X)$

$X$ is a random variable. Its distribution function and characteristic function are $F_X$ and $\phi_X$, respectively. Then, we know, $F_X(X)$ follows uniform distribution. Let's say, $U=F_X(X)$. My ...
2
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1answer
96 views

Product of Uniform random variable

Given three iid random variables, $X,Y,Z$ following a Uniform $(0,1)$ distribution. Then, how to find $\Pr(X>YZ)$? Recently, I was asked this question in an interview, but I chocked there. ...
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0answers
18 views

Distribution of transformed poisson distribution

Let X follows Poisson distribution with parameter $\lambda\gt0$ and $Y=aX$, where $a\gt0$ is a constant. Q. What will be the $PMF$ of $Y$? Since by using the M.G.F we have ...
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2answers
42 views

joint probability and conditional probability question

The number of workplace injuries, $N$, occuring in a factory on any given day is Poisson distributed with mean $\lambda$ . The parameter $\lambda$ is a random variable that is determined by the level ...
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1answer
33 views

Let $X~Bin(n, \lambda/n)$, $\lambda >0$. Show that for fixed $k \geq 0$, $P(X=k)\equiv \frac{e^-\lambda)\lambda^k}{k!}$

Let $X - Bin(n, \lambda/n)$, $\lambda >0$. By using approximation $(1-\frac{x}{n})^n\approx e^{-x}$. Show that for fixed $k\geq 0$, $P(X=k)\approx \frac{e^{-\lambda}\lambda^k}{k!}$ ...
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0answers
47 views

conditional density of $X$ given $XY^2$

I was asked the following problem. Given that $X$ and $Y$ are random variables with joint density $f(x,y)$, find the conditional density of $X$ given $XY^2$. My thought was to first change variables ...
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1answer
64 views

Computing Conditional Variance

I have been tasked with trying to solve a conditional variance. I have red and black pens with respective exponential probability parameters 2 and 4. I have 70% red pens and 30% black pens. What is ...
2
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1answer
43 views

$X \sim Rice(\nu,\sigma)$, what is the distirbution of $X^2$?

Let $X = |\nu e^{j\theta}+W|$, where $W \sim \mathcal{CN}(0,2\sigma^2)$, i.e. $X\sim Rice(\nu,\sigma)$, what is the distirbution of $X^2$? Note that X also can be writen in terms of real and imaginary ...
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1answer
100 views

Convolution of Two Random Variables

I have been working a few hours on this particular problem. Please excuse my lack of formatting. This is the question: Let $X$ and $Y$ be random variables with density function $f(x) = 2x$ on $[0, ...
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1answer
18 views

Metropolis-Hastings Algorithm Clarification

All- Could you please clarify: from wikipedia, step two states at the end if the candidate is rejected, set xt+1 = xt, instead. I don't quite understand this, so you will have two of the exact same ...
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1answer
25 views

Did I do this Continuous Probability Problem Correctly?

I'm new to evaluating continuous probability density functions. I'd like someone to check my work, please. Problem: Suppose $X$ has density $f(x) = c/x^6$ for $x>1$ and $f(x) = 0$ otherwise, ...
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1answer
27 views

Distribution of sine composed with a random variable

Could you tell me if my calculations are correct? We are given a random variable with the following discrete distribution $$P(X=n) = \frac{2^n}{3^{n+1}}, \ \ n \in \mathbb{N}.$$ Find the ...
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0answers
46 views

Prove the Fourier Inversion Formula for a Multivariate Distribution

Question: Prove the Fourier Inversion Formula for the specific function $\phi_{\Sigma, \mu}(x)$: $$\phi_{\Sigma, \mu}(x) = (2\pi)^{-k} \int_{R^k}\hat{\phi}_{\Sigma,\mu}(\xi)e^{-i\xi\cdot x}d\xi$$ ...
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1answer
64 views

Ratio of two beta random variables

I'm working on a problem for an hour and I wanted to get some hints. Suppose: $y_1, y_2, y_3, y_4 \sim Dir(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$ what is the distribution of $\frac{y_1}{y_1 + ...
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1answer
52 views

Convergence in Distribution and Exponential Function

There's a well known fact that if a sequence of real numbers, $\{x_{n}\}$ converges to $x$, then: \begin{equation*} \lim\limits_{n\rightarrow\infty}\left(1+\dfrac{x_{n}}{n}\right)^{n} ...
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1answer
40 views

cdfs $F$ and $G$ of random variable $X$, $F\le G$. What can we say about $\mathbb{E}_F[X]$ and $\mathbb{E}_G[X]$?

Problem: A random variable $X$ is distributed in $[0,1]$. Mr. Fox believes that $X$ follows a distribution with cumulative density function $F:[0,1]\to [0,1]$ and Mr. Goat believes that $X$ follows a ...
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1answer
20 views

$X$ and $Y$ are unformly distributed in $[0,1]$ with $P(\max(X,Y)≤z)=P(\min(X,Y)≤(1−z))$. Find $z$.

Problem: Two independent random variables $X$ and $Y$ are uniformly distributed in the interval $[0,1]$. For a $z \in [0,1]$, we are told that $P(\max(X,Y)\le z)=P(\min(X,Y)\le (1-z))$. Then, what is ...
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1answer
89 views

Exponential Probability Question

A college buys 70% of dorm light bulbs from Company A with an exponential lifetime $f_A(x)~ exp(\lambda = 2)$. The other 30% come from company B have lifetime $f_B(x) ~exp(\lambda = 4)$. At the start ...
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1answer
44 views

The sum of infinitely many independent Poisson random variables.

I'll post my own answer to this unless someone beats me to it and maybe even after ten others are posted in the first ten minutes, but of course there may be many ways to prove the result, so post ...
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1answer
62 views

What kind of distribution is this? Use Moment Generating functions

Let X Pois($\phi$) and Y Pois($\tau$) be independent poisson random variables. a) Use moment generating functions to show that Z = X + Y Pois($\phi +\tau$ ) b) Find the conditional distribution of X ...
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1answer
82 views

An application of the Central Limit Theorem

Suppose $X_i$ are independent random variables uniformly distributed on $[1,3]$. We are interested in the product $W=X_1X_2\cdots X_{10}$. Each $X_i$ is centered about $2$ so we might think $W$ should ...
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1answer
38 views

Pattern Recognizing in Integrals and Probability Distirbutions

This is a basic pattern recognizing question to begin with, but asks about probability densities. Solve the following integrals and find a pattern. I was able to solve them all and they are all ...
2
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1answer
64 views

Marginal Densities

I just have a few questions about joint density and marginal density questions. Q1: Joint Distribution $f_1=2x+4y$ on triangle with vertices $(0,0), (0,1),(1,0)$. Sketch the region and compute ...
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0answers
41 views

Coalescent theory - Why are coalescent times independent?

I am reading from this book and I want to make sure I understand what is going on. What I get from the book Consider a population of $N$ individuals. The population size ($N$) is constant. select ...
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25 views

Dsitribution of $|Ae^{j\phi} + W(t)|$, where $\phi \sim unif[-\pi,\pi]$

Let $Y(t) = Ae^{j\phi} + W(t)$, where $\phi \sim unif[-\pi,\pi]$ and $W \sim \mathcal{N}(0,\sigma^2)$. What is the probability distribution of $|Y(t)|$ ? If $\phi$ was deterministic, i.e. a constant ...