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

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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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0answers
3 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
7 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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1answer
18 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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2answers
36 views

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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2answers
171 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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1answer
15 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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0answers
6 views

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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1answer
28 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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1answer
36 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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1answer
11 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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1answer
43 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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1answer
19 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 ...
4
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1answer
74 views

Is there any short proof of this classical problem?

Let $X,Y$ be two i.i.d. r.v.'s with zero mean and unit variance. If $X+Y$ and $X-Y$ are independent, then $X$ and $Y$ are both standard normal distributed. Is there any short proof for this problem?
4
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0answers
27 views

shifted exponential distribution with inter-arrival time

Given that time interval $T^*$ in seconds between certain events has a negative exponential distribution. The instrument cannot detect intervals which are less than $\delta$ seconds. Let $T_1, ..., ...
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1answer
31 views

probability about animals being moved

I have this scenario: 1 animal with 30% probability of be moved to Japan. 1 animal with 30% probability of be moved to Japan. 1 animal with 30% probability of be moved to Japan. 1 animal ...
2
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1answer
18 views

Central Limit problem

The times that patients spend in a doctor’s surgery have mean 5 minutes, and standard deviation 2 minutes. On one particular day, the doctor sees 30 patients during his surgery which starts at 4.30pm. ...
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1answer
40 views

On intervals chosen randomly within the unit circle.

Let $S = \{(x,y)\in R^2 : x^2 + y^2 = 1\}$ be the unit circle in $R^2$. Let $(X_1, Y_1), (X_2, Y_2)$ be independent, both having uniform distribution over $S$. Let $D$ denote the Euclidean distance ...
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2answers
70 views

Proving the sum of two independent Cauchy Random Variables is Cauchy

Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt ...
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0answers
19 views

Distribution of continuous time markov chain

I'm having trouble understanding the question below. I understand the continuous time markov chain and unique stationary distribution but not sure what it is asking. I have a continuous-time Markov ...
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0answers
8 views

Calculation c.d.f or p.d.f with max function

Provided $\{\lambda_i\mid i =1,2,\cdots,12\}$, where each $\lambda_i$ is i.i.d with exponential distributed with rate parameter 1. Define $\lambda_{max}=\max\{\lambda_i\mid i =1,2,\cdots,12\}$. ...
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0answers
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Distribution of sample statistics taken from bivariate normal

$(X_{1},Y_{1}),\,...\,,(X_{n},Y_{n})' (n>2)$ are random samples taken from $N_{2}((\mu_{1},\,\mu_{2})',\,$$ \begin{pmatrix} \sigma^{2}_{1} & \rho\sigma_{1}\sigma_{2} \\ ...
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26 views

lottery Probability Estimation [on hold]

A sales manager is keen to find out the optimal order quantity for a certain product, and gives the judgement that he is indifferent between two lotteries when demand of the product and probability ...
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2answers
47 views

If $X$ is standard Normal then find $\lim_{x\to0}P(X>x+\frac{a}{x}|X>x)$

If $X$ is Standard Normal and $a>0$ is a constant then find $\lim_{x\to0}P\big(X>x+\dfrac{a}{x}\big|X>x\big)$. This is an exercise from a book whose name I cannot immediately recall. I ...
2
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2answers
22 views

how to understand the generation of cauchy distribution from uniform distribution?

I am learning some basic idea on generating cauchy distribution from uniform random generator $u \in [0, 1]$. I know it was discussed before in How to generate a Cauchy random variable, but during my ...
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0answers
28 views

Probability Generating Function of a Negative Multinomial Distribution

Derive the probability generating function (pfg) of a negative multinomial distribution with parameters $(k; p_{0}, p_{1}, ..., p_{r})$ where the k-th occurrence of the event with the probability ...
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0answers
14 views

limit of characteristic function Normal

I need to find the limit of the following characteristic function as $s \rightarrow\infty$ $\frac{e^{-it\frac{s}{\sqrt{s^2+s}}}}{(1-(e^{-it\frac{1}{\sqrt{s^2+s}}}-1)s)}$ The top part seems to reduce ...
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3answers
31 views

Probability of balls drawn with replacement

We have two bags, Bag A has 40 red balls and 15 blue balls, Bag B has 40 blue balls and 10 red balls. One of these bags is selected at random and from it five balls are drawn at random, replacing each ...
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1answer
30 views

Proving the variance of pareto random variable equals (a*lambda)/((a-1)^2*(a-2))

So my PDF for the Pareto distribution is: $$\dfrac{a\lambda^a}{x^{a+1}},\quad x\ge\lambda$$ To find the variance, you need to find the integral of $x^2\dfrac{a\lambda^a}{x^{a+1}}$ and subtract it from ...
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1answer
13 views

Visualizing a probability measures through a probability density functions

I found a previous question with a very nice answer, but still there is something that is not completely clear to me. We start from a space $(X, \Sigma)$, endowed with a $\sigma$-algebra, and we let ...
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0answers
20 views

Distribution of some linear combination of Normal RVs

I would like to ask for help concerning this question lifted from the book An Introduction to the Theory of Statistics by Mood, Graybill, and Boes (2nd ed.). Let $X_1$ and $X_2$ be independent ...
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3answers
46 views

Finding a constant $ z $ such that $ P(Z \leq z) = 0.95 $ when $ Z \sim \text{N}(0,1) $.

This is for a homework assignment on normal distributions. Question: a) Find a constant $z$ such that $P(Z \leq z) = 0.95$ b) Find a constant $z$ such that $P(Z \geq z) = 0.95$ I'm having trouble ...
0
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1answer
18 views

Sum of multiple distributions

Three fish $F_1, F_2, F_3 $ are selected at random from the pond, their weights are independent and identically distributed $ \sim N(20.3, 0.13^2) $ a) Find $ P(\bar{F} \ge 20.1) $ b) Find $ P(F_1 ...
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2answers
35 views

How to calculate the probability distribution function (PDF)?

Sorry for the dumb question, I've been struggling with understanding the probability distribution function formula, what does "x" and "d" stand for in the formula , and how to use the formula? I've ...
1
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1answer
15 views

Compute lower tail probability from upper tail - bivariate normal

Let $X,Y$ be bivariate normal with correlation $\rho$. I'd like to compute $\mathbb{P}(X \leq x, Y \leq y)$, but I have only a function (implemented on a computer) that returns $\mathbb{P}(X \geq x, ...
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1answer
19 views

Birth-death process: What is the distribution of reached states before reaching an absorbing state?

Intro I am working on a birth-death process. For a given choice of parameter ($n=6$, $Wa=1$, $Wb=0.95$, see below), the transition matrix is $$\left( \begin{array}{ccccccc} 1. & 0.144928 & ...
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0answers
24 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 ...
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0answers
27 views

what is determinantal process?

Would anyone please explain what does this mean? A random point process $P$ on a discrete base set $Y = \{1,\ldots,N\}$ is a probability measure on the set $2^Y$ of all subsets of $Y$. Let $K$ ...
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0answers
12 views

A question about the translation property Markov kernel

Given that ${X_n}$ is a Markov chain, and a Markov kernel with translation propert$p(y+x,E+x)=p(y,E)$. Question:How to show $Y_n=X_n-X_{n-1}$ are i.i.d? I'm trying to use Markov Property and ...
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0answers
8 views

Normalizing constant for product of Gaussian densities - interpretation

The normalizing "constant" for the product of two multivariate Gaussian densities, with mean vectors $a$ and $b$ respectively, and covariance matrices $A$ and $B$ respectively, is (the reciproke of) a ...
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1answer
18 views

How to determine the support (bounds) of a cumulative distribution function

Suppose that X is uniformly distributed on [0,2]. Suppose that Y = X$^3$ Find the probability density function for Y and state the support for Y. I know the CDF will be G(y) = P(Y $\le$ y) = P(X$^3$ ...
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1answer
30 views

An unbiased estimator for the parameter of exponential distribution

The times between arrivals at a fish shop queue can be described by an exponential distribution $X$ with parameter $\theta$ . If $Y$ is the random variable given by the sample mean on $n$ ...
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1answer
36 views

2 User Queuing Model Probability Problem

Consider two users who arrive to a system with exponential arrival times with parameters $\lambda_a$ and $\lambda_b$. Once they arrive, the users stay in the system for an exponentially distributed ...
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0answers
29 views

Random variables and the topology of weak convergence

To see what's going on, I am trying to translate the idea of topology of weak convergence on a random variable setting, just to get some concrete intuition. This is what I have got so far (where the ...
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0answers
21 views

What does it mean “rotationally invariant density”?

In the great answer given by the math.SE user @Tim, he does 2 hypothesis, on of the which ones is about the rotationally invariance of the density. Can you explain formally what does it mean? I do ...
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1answer
23 views

continuous probability density functions

Continuous distributions assign probability 0 to individual values. But, according to DeGroot, it doesn't mean that it is impossible for the random variable to take individual values. So, why not make ...
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1answer
43 views

Proving conditional distributions are normal [on hold]

For the standard bivariate normal distribution, it is easy to show (by simple integration) that both marginal distributions are N(0, 1). Prove that both of the conditional distributions are also ...
5
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1answer
39 views

Distribution of occurrences of “pairs of heads” in $N$ coin tosses

Let's say we toss a weighted coin $N$ times, each with probability $p$ of landing heads up. What's the distribution of the number of times we'll see $k$ pairs of heads? For example, HTHHHTHH would ...
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2answers
63 views

Binomial Random Variable and Bernoulli trials problem

Let X be a Binomial random variable defined as the sum of 6 independent Bernoulli trials. The probability of a Bernoulli taking the value 1 is given by p. Suppose that prior to the 6 Bernoulli trials, ...
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1answer
29 views

Joint Probability with Poisson distribution [on hold]

Let $X$ and $Y$ be independent Poisson random variables with means 9 and 16, respectively. Compute (a) $E[\min\{X, 2\}]$, (b) $\text{Var}(X + Y)$, (c) $\text{Cov}(X + Y, X)$.