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

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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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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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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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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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37 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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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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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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52 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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68 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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34 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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38 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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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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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
29 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
39 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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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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65 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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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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23 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
83 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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23 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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86 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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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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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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35 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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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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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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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
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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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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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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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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 ...
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PDF of $|X(t)| =| e^{j\omega_c t}+W(t)|$

let $X(t) = Ae^{j\omega_c t}+W(t)$, where $W(t)$ is a gaussian process that follows the statistics $W \sim \mathcal{CN}(0,\sigma^2)$ and $\omega_c$ denotes the carrier pulse frequency and $A$ is a ...
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1answer
22 views

Exponential Distribution Unbiased Estimate of Coefficient of Variation?

Through simulation, I've noticed that estimates of the coefficient of variation (CV) of exponentially distributed variables are biased at low sample sizes (as seen in the plot I made). I've seen an ...
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1answer
14 views

Issue with sum of probabilities of probability distribution function of a geometric random variable

Is it possible that the sum of probabilities of geometric distribution for "$k = 1,...,n$", where k is number of trials until the first success, is not equal to 1? I'm asking this, because I encounter ...
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27 views

Distribution of the sum of random variables

Let $X_{1}$,$X_{2}$,...,$X_{N}$ be a Dirac distributed (not independent) random variables. What is the distribution of $\sum_{i=1}^{N}{X_{i}}$?
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Given the probability distribution of X, whats the PDF of X²?

Let's say we have a random variable $X$ with a certain probability density function $f_x(x)$. 1) How should I find out the PDF of the random variable $X^2$? Problem background: $X_1 = s_1 + W$, ...
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186 views

Convergence in Probability

Consider a sequence of $N$ Bernoulli trials with, with probability of success denoted by $p$, and let $X$ be the number of successes. Show that as $N\rightarrow\infty$, $\frac{X}{N}$ converges in ...
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21 views

Probability function of Acos(x)

Let's say I have a signal $y(t) = Acos(2\pi f_c t)$, where $f_c$ is the carrier frequency and $t$ is the independent variable. Since I work with discrete signals i sample this signal with a sampling ...
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Example of Semi Markov Process, that isn't a Markov Chain in Continuous Time?

Question says it all I hope. I have an exam in Stochastic Processes tomorrow and one question that may be asked is to give an example of a Semi-Markov Process that isn't a Markov Chain in Continuous ...
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42 views

To show $X$ and $|X|$ are not jointly continuous

Suppose $X\in N(0,1)$. Show that $X$ and $|X|$ are not jointly continuous. I am not sure how I can approach this problem. But the following method seems plausible to me: $$P(X\leq ...
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58 views

Poisson Process. Expected time of three fishermen catching at least three fish.

Three fishermen are fishing, we model the fishing as a Poisson Process of rate $2.5$ fish/hour. The fishermen leave only when each of them them has caught at least 3 fish, we call this leaving time ...
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18 views

composition of probability distribution functions

Suppose we are given $X \sim \mathcal{N}(\mu,\Sigma)$. Then, we define the random variable $Y$ as follows: $Y_i = 1 + X_i $ if $X_i \ge 0$ $Y_i = \exp(X_i)$ if $X_i \lt 0$. How do I go about ...
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54 views

Probability question for economics that I'm struggling with. Please help.

(There are 4 districts in the land of Oz. At home, the inhabitants of each region wear ties of a special colour, Munchkins (M) wear blue, Scarecrows (S) wear purple, Tin Men (T) wear red and Wizards ...
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30 views

A question on joint probability density functions.

I know that the pdf $X$ conditional on $Y=y$ is $$f_{X|Y}(x|y)=\frac{f_{(X,Y)}(x,y)}{f_Y(y)},$$ and this can be used to calculate conditional probabilities such as $P(X>\alpha | Y>\beta)$ (for ...