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

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20 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
23 views

explanation : Example of Gaussian random process?

Can any one explain to me how to answer the question and what is the Gaussian random process in a simple way. I know how we find the C xx from R xx the rest of the answer I don't understand why all ...
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0answers
24 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
47 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 ...
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1answer
35 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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0answers
26 views

independence copula diagonal

I'm reading Nelsen's Instruduction to copulas, and there is (probably very simple) excersice I cannot deal with. It says that if the diagonal section of the copula equals the diagonal of independence ...
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1answer
41 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
12 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
24 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
10 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
17 views

X correlates with Y and Y correlates with Z when p-values are known

"How can I calculate the range of correlation of two variables X and Z given I have the correlations of X and Y, and Y and Z?" These are useful resources: ...
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0answers
18 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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0answers
16 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
24 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
25 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
17 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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0answers
11 views

Multivariate distribution expectation

For this question I found that Pmf of X: 2/5 when x=0, 3/10 when x=1, and 3/10 when x=2 For the Pmf of Y: 3/5 when x=0 and 2/5 when x=1 But I got confused on part c, I am not sure how to calculate ...
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0answers
36 views

Differential equation whose solution is Erlang distribution

I am working on a proof (Probability Density Question Involving an Integral Equation (from Karlin & Taylor's A First Course on Stochastic Processes)) and got stuck. Now I would like try ...
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1answer
66 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
21 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
51 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
66 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
33 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 ...
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1answer
37 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
28 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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0answers
17 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 ...
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1answer
26 views

Distribution of the sum of squared independent normal random variables

How do I go about finding the the pdf of the statisitc $\sum_i x_i ^2$ such that each $x_i$ is iid from a $N(\sigma , \sigma)$ distribution? I've searched, but cannot find a straightforward answer. ...
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1answer
28 views

Characteristic function of a product of two dependent random variables such that one is continuous the other is discreet

If you're given the characteristic function of a continuous random variable, say $X$, and the distribution of another discreet random variable, say $U$, which is dependent of $X$, how do you ...
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2answers
36 views

Joint Density of N Dependent Uniformly Distributed Random Variables

Could someone show me the formula with proof for the Joint Density and CDF for N uniformly distributed variables that are not necessarily independent? Again, if certain forms of dependence are ...
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0answers
26 views

Decisions with Probabilities - application

A company has to decide whether to manufacture a product at its plant or purchase from a supplier. The resulting profit depends on the demand for the product. The estimated profit is shown below: ...
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1answer
61 views

Coin Toss Question with Normal Distribution.

Toss a coin and if it lands on heads, then X is distributed normal with Mean=1, Variance=1. If it lands on tails, X is distributed normal with Mean=-1, Variance=1. (a) Given X=1, what is the chance ...
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0answers
13 views

Probability impact of portfolio size increase

We are defining a new Venture Capital portfolio size (# of investments) in the light of the Power law (20% of the investments will pay for the whole invested amount and generate the returns). We ...
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0answers
22 views

Uniform most powerful Test for one-sided hypothesis

I am trying to understand this proof above. What I am confusing is (1) The whole theorem correspond to the hypothesis $H_0:θ\leθ_0 \, vs \, H_1:θ\gtθ_0$. But at the beginning of the ...
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4answers
82 views

Find density function of $X + Y$ , where $X, Y$ random variables.

Here's a question: Let $(X, Y)$ be random vector that distributes uniformly in the triangle with coordinates: $(0, 0) , (1, 0), (0, 1).$ Question: Let $Z = X + Y$. Calculate density function ...
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2answers
21 views

Chebyshev Probability question

I'm looking at a question that says if you have a random variable $X$ with mean $\mu$ and variance $\sigma^2$ and $\sigma=0$ then prove that $X=\mu$ with probability 1. I'm pretty sure it has to be ...
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2answers
22 views

Poisson process. Patients arriving te the ER.

People arrive to the ER of a hospital following a poisson process with $\lambda=2.1$ patients/hour. One of each 28 who arrives under this condition, dies. Calculate the probability of: (a) At least ...
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1answer
9 views

Product distribution function of two independent (mixture of discreet and continuous) random Variables

Consider two random variables say X and Y, the first being the standard normal, while the other is the radamecher taking a value of 1 or -1 each with probability 0.5. what is the probability density ...
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2answers
85 views

Is there a probability distribution with mean $1$ such that $f(x)=\frac{1}{x}f\left(\frac{1}{x}\right)$

Is there a probability distribution defined over $\mathbb{R}^{+}$ by the pdf $f$ such that, $$\forall x > 0, f(x)=\frac{1}{x}f\left(\frac{1}{x}\right)$$ and $$\int_0^{\infty} x~\mathrm{d}f = 1 $$ ...
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2answers
46 views

Variance $= 0$, show that $X=\mu$ with probability one

If the variance of $X$ is zero, show that $X=\mu$ with probability one. Using Chebychev's inequality that is, \begin{equation*} P(|X-\mu|\geq k\sigma)\leq\frac{1}{k^2}, \end{equation*} I just let ...
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0answers
15 views

Sum of folded normal distributions

Set three random variables \begin{equation*} X_1, X_2, X_3 \sim \mathcal{N}(\mu, \sigma) \end{equation*} and their respective transformations \begin{equation*} Y_1 = |X_1| \;,\; Y_2=|X_2| \;,\; ...
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1answer
25 views

Characteristic function of a product of two dependent random variables

If you're given the characteristic function of a continuous random variable, say X, and the distribution of another discreet random variable, say U, which is dependent of X, how do you explicitly find ...
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1answer
8 views

Probability measure and distribution

What is the difference between these two terms? From what I saw the two terms are not exactly interchangeable. I think that probability distribution implies probability measure, but the converse is ...
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1answer
31 views

Find$P(X>Y^2)$ for $f(x,y)$

Find $P(X>Y^2)$ for $f(x,y)=9x^2y^2$ where $0<y<1$ and $-y<x<y$ I know that $P(Y^2<X)=P(-\sqrt{X}<Y<\sqrt{X})$. Is this useful?
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1answer
41 views

Distribution of function of a Random Variable

If $X$ is uniform on $(0,1)$, how would I go about finding the CDF of $Y=(X-X^2)^2$ ?Thanks.
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1answer
35 views

Definition of the conditional pdf

I was wondering how the definition of the conditional pdf came about where, $$f_{(X,Y)}(x|y) = \frac{f_{(X,Y)}(x,y)}{f_Y(y)}$$ I'm trying to understand this by thinking of the conditional CDF of ...
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1answer
22 views

Combination of two bivariate Gaussian covariance matrices

I have measurements of 2 position vectors ($\mathbf p_1$ and $\mathbf p_2$): Each with their own mean position vectors $(\overline x_1, \overline y_1, \overline z_1)^T$ and $(\overline x_2,\overline ...
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1answer
13 views

U uniform on [-1,1] - Find density of U^2

Let $U$ be uniformly distributed on $[-1,1]$. Find the denstiy of $U^2$. I would start with $F_{U^2}(u)$=$P(U^2\le u)$=$P(-\sqrt{u}\le U\le\sqrt{u})$ for $u\ge 0$. Since it is uniformly distributed ...
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24 views

distribution of quadratic form of jointly normal random variables?

I need to derive the distribution of the random variable $\frac{W'(I-1(1'1)^{-1}1')ZZ'(I-1(1'1)^{-1}1')W} {Z'(I-1(1'1)^{-1}1')Z}$ , where $(Z, W)'$ ~ $N(0, I), \,Z=(Z_{1}, ..., Z_{n}), \,W=(W_{1}, ...
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2answers
25 views

Compute variance of logistic distribution

Consider a random variable $X$ with normalized logistic distribution( so that its pdf is $\frac{e^{-x}}{(1+e^{-x})^2}$). It is well known that its variance $V$ equals $\frac{\pi^2}{3}$ but I couldn't ...
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1answer
27 views

Density of a compound poisson process.

People arrive to a bank according to a possion process $N(t)$ with $\lambda = 1$ client/minute. Each client makes a deposit $Y \sim \mathrm{Unif}\{1,2\}$ in thousand dollars. Calculate the probability ...