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

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33 views

Probability distribution of request handling

I have values representing time taken to execute one request on server. Could somebody advise what type of distribution it is? I think that normal distribution but I am not really sure about it. ...
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1answer
33 views

Dependent Bernoulli trials confidence interval

I would like to know if there is a way to build a confidence interval, for a random variable which has a Bernoulli distribution, based on its history. I mean if the order of its states is 11100 (i.e. ...
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28 views

how to determine transient and recurrent state from transition matrix

I wonder how can I determine the transient and recurrent state from transition matrix ? I mean if I have 10 states It would be very hard to draw diagram for them so how to analyse the matrix? For ...
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0answers
13 views

For which values of $ a $ can this function be a valid distribution function?

Let $ F(x) = a(x+1)^2(u(x+1)-u(x-1))+u(x+1) $ . For which values of $ a $ can this function be a valid distribution function? I couldn't solve this question. Because for $x \geq 1 \implies F(x) = ...
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1answer
21 views

CDF of two variable

I would like to calculate the CDF of sum of two random variable in a unit square I realize that everywhere says if X+Y=z and then if z is between 0 and 1 then probability is equal to something and if ...
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1answer
19 views

Compound poisson process [closed]

Let {$X(t): t \geq 0$} be independent Poisson processes with respective parameters $\lambda$ and $\mu$. For a fixed integer $a$, let $T_a$ = min{$t \geq 0; Y(t) = a$} be the random time that the Y ...
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1answer
16 views

probability of joint PDF

I found $k = 4$ and yes, the are independent. But for the last one I know how to find the probability if they are like $x$ from $0$ to a number and $y$ from $0$ to a number so the limit of double ...
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0answers
35 views

Pdf of variable as combination of two random variables with exponential distribution

If $X$ and $Y$ are independent and exponentially distributed, which is the pdf of $Z$? Where $Z$ is given by \begin{equation} Z = \frac{X}{1+Y} \end{equation} I read answer to this post: $X,Y$ are ...
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1answer
16 views

Representation of a non-standard normal variable squared

I have come across a representation of a non-standard normal distributed variable square. It is clear for me that assuming $Z_j \approx N\left ( \theta_j, \frac{\sigma^2}{n} \right )$ we can write ...
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1answer
21 views

Calculating Chi-square probability with $X^2$ and degrees of freedom

How do I calculate the probability (%) for chi square test using $X^2$ value and DoF as inputs? Im trying to create a C++ program to calculate chi square tests with very high DoF, so I cannot use the ...
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22 views

Proving a reflection principle in probability [closed]

This is a problem that I am stuck at. I tried to prove these exercises by dividing cases, but it only seems to complicate the matter. Could anyone help me how to solve this problem?
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0answers
19 views

inhomogeneous poisson with exponential lambda function

Suppose that a random variable X is distributed according to a Poisson distribution with parameter $\lambda$. The parameter $\lambda$ itself is a random variable, exponentially distributed with ...
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1answer
23 views

How were the bounds of integration changed here in this probability problem on uniform distributions?

I have the following problem and solution: I don't understand how the bounds of integration were changed from 0 to 1, to $x^2$ to 1. I see where $1/\sqrt{y}$ was substituted in for $f(x|y)$ and 1 ...
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0answers
14 views

Is it true that $X(t) \sim Pois(p\lambda) = Pois\left(\frac{\lambda_1}{\lambda_1 + \lambda_2} \cdot \lambda\right)$ [closed]

If: $$N(t) \sim Pois(\lambda)$$ $$X(t) \sim Pois(\lambda_1)$$ $$Y(t) \sim Pois(\lambda_2)$$ $$N(t) = X(t) + Y(t) \sim Pois(\lambda_1 + \lambda_2) = Pois(\lambda)$$ Then is it true that $$X(t) \sim ...
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1answer
27 views

Simulating r.v.'s from a joint density by using rejection sampling in R

I wish to sample variables $v$ and $w$ from the joint density $$(v+w)e^{-\frac{(v+w)^{2}}{2x_{0}}-2\mu v-(\mu -\lambda )w},$$ where $x_0$, $\mu$ and $\lambda$ can be seen as positive constant. Since ...
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1answer
27 views

Uniform Distribution Probability QUestion [closed]

Let $(X, Y )$ have a uniform distribution on the set $$\{(x, y) : 0 < x < 2 \text{ and } 0 < y < 8 \text{ and } x < y\}$$ Find $P(Y < X^3)$
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1answer
20 views

Relation between vague convergence and weak convergence

This is the Portemanteau Theorem. And this is its corollary. I tried to prove that (i) implies (ii) in this corollary using the Portemanteau Theorem above. But I have kept failed... What is so ...
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1answer
40 views

Find the constant value in the Probability mass function

I have a Random Variable N that has a pmf: $$p(N=k)=C(k+1)(n-k+1), \text{ for } k=0,1,2,..., n \text{ and } C \text{ is a constant }$$ I am asked to find (a) the constant value $C$, (b) the CDF and ...
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1answer
28 views

Kullback-Leibler divergence when the $Q$ distribution has zero values

For discrete probability distributions $P,Q$, the Kullback-Leibler divergence of $Q$ from $P$ is defined to be $$D_{\mathrm{KL}} ( P \mathop{\|} Q ) = \sum_i P(i) \ln \left( \frac{P(i)}{Q(i)} ...
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18 views

Difference of chi-squared distributions

This question seems simple, but it gets me a little confused. Suppose that $U$ and $V$ are independent and $U\sim \chi_k^2$ and $W=U+V \sim \chi_{k+l}^2$. Can we then immediately say that $V\sim ...
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1answer
20 views

How to extract rates and expectations from a poisson distribution.

I'm reading this wikipedia article to try to get an understanding of the Poisson distribution (again). From what I'm reading, it seems like the Poisson distribution is useful for answering the ...
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2answers
12 views

Uniqueness of probability given marginals

Let $X,Y,Z$ be finite sets, and consider probability distributions $p$ over $X\times Y\times Z$. If we know the marginals of $p$ over all the pairs $X\times Y$, $X\times Z$ and $Y\times Z$, is that ...
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1answer
31 views

Campbell's theorem variance

From Wikipedia, For a Poisson point process $N$ and a measurable function $f: \textbf{R}^d\rightarrow \textbf{R}$, the random sum $$\Sigma=\sum_{x\in {N}}f(x)$$ [...for complex value ...
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0answers
21 views

Gaussian distribution finite population with unknown cardinality

I have taken a sample population of a population with unknown size. The sample size is 54 trades. The sample mean is 2.1% (1.021) return per trade. The sample standard deviation is 0.01. 100% of ...
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38 views

Sum of number of rows with max value

Suppose i have an N by N matrix, each element in the matrix my contains 0 or 1, so there are 2^(N*N) different matrix. Let's define the function F that takes a matrix and calculate the sum for each ...
2
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0answers
59 views

Name of a “factorial” distribution

Is there a common name for the distribution $P(m)=\frac{(m-1)}{m!}$, for integers $m\in \{1,\infty\}$? Its mean is $e$.
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0answers
11 views

In a normal distribution curve, why is the probability of Z being greater than 1.64 bigger than Z being greater than 0? [closed]

I've found the probability of a point Z in a normal distribution diagram being greater than 1.64 is 0.505 and the probability of Z being bigger than 0 is 0.5. if you look at it, the probability of Z ...
2
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2answers
43 views

Variance of the Maximum Likelihood Estimator of the parameter of a Rayleigh distribution

I want to calculate the variance of the maximum likelihood estimator of a Rayleigh distribution using $N$ observations. The density probability function of this distribution is : $$ f(\sigma,y_i) = ...
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1answer
29 views

Differentiate joint cumulative distribution function

I have probability $P(X-a>Y)$ and I get CDF function as following (no explicit PDF form was given), and $F(a)$and $f(x,y)$ are CDF and PDF funtions, respectively. ...
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1answer
55 views

Variance of the number of r.v summed to fill certain capacity

Let us assume that we have a certain capacity T. We have an infinite number of random variables $X_1,X_2,\dots,$ where each $X_i$ is independent and has a particular pdf $P_i(X)$. And we have that ...
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1answer
7 views

Cumulative distribution of first success in bernoulli trials with p = 0.5

A box contains a very large number of balls, so that the probability of choosing a white or red (initially at equal numbers) remains at 1/2 as balls are chosen. Let X be the number of balls chosen at ...
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0answers
18 views

How do I find the PD.F for the function of two random variables?

I am trying to find the P.D.F for the equation below that is a function of two random variables; i = $f (\sqrt E_{p}. e^{-(t-\mu)^2/(2\sigma^2)})$ How do I find the P.D.F for this equation where ...
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0answers
10 views

Find the mean of lognormal rv's with available variance and the sum of rv's

I have the sum of a bunch of random variables $S$, v = [1 1 2 2 3 3 4 ...]; S = sum(v); I know that vector $v$ is lognormally distributed, BUT I DON'T KNOW IT. ...
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0answers
8 views

Inverse of a conditional CDF involving Copula

I am planning a Monte-Carlo simulation exercise involving Gaussian Copulae. I have $n$ random variables $X_1,X_2,X_3,...,X_n$ with known CDFs $F_i(x)$ (the CDFs are known but can not be described by ...
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0answers
9 views

Hypothesis Question

Consider the following hypothesis: $H_{0}:\mu\leq3000$ vs $H_{a}:\mu>3000$ A sample size $n$ must be decided so the risk of a type 1 error is at most 1%, and also so that if the value of ...
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1answer
136 views

uniformly distributed random variables

Ram and Shyam wanted to meet at a park about 12.30 P.M.. If Ram arrives at a time uniformly distributed between 12.15 P.M. to 12.45 P.M. and if Shyam independently arrives at a time uniformly ...
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2answers
27 views

What is the distribution given by $\int^t_0 W_s^2ds$

Define $X_t=\int^t_0 W_s^2ds$, what will be the distribution of $X_t$? My approach is as follow: Let $f(s)=W_s^2s$, by Ito's lemma we have $X_t=W_t^2t-2\int^t_0W_ssdW_s-\frac{t^2}{2}$. Discretize ...
2
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1answer
60 views

Probability of machine being adjusted unnecessarily

In my math book I found the following problem: Some company bottles a liquid in bottles of X ml, where X is distributed normally. $\mu_X = 400$ and $\sigma_X = 4$. (I'm not sure of the ...
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1answer
17 views

Checking if $Z$ is an F-distribution using change of variable technique

Two independent random variables $X_1$ and $X_2$ have the following pdf $f(x_1,x_2)= e^{-x_1-x_2}$ for $x_1, x_2>0$, and $0$, otherwise. Using the change of variable technique, determine whether ...
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0answers
18 views

Joint p.d.f $Y=x_1/x_2$ for two independent continuous random variables $X_1$ and $X_2$

The question reads like this: Two independent continuous random variables $X_1$ and $X_2$ have a joint p.d.f $f(x_1,x_2)$. Determine the p.d.f of $Y=X_2/X_1$, assuming $Y>0$. (That is $Y$ is ...
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1answer
44 views

Probability of a sample mean for a bivariate probability density function

The bivariate probability density function for two random variables $X$ and $Y$ equals the following: $f(x,y)=12x^2y^3$ for $0<x<1$ and $0<y<1$; $f(x,y)=0$ otherwise. $X$ and $Y$ are ...
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1answer
21 views

Calculating expected value of a pareto distribution

Suppose that you have a Pareto product distribution function defined by: $$ f(x;k;\theta)= \begin{cases} \frac{k\theta^k}{x^{k+1}} & x \ge\theta \\ 0 & x \lt \theta \end{cases} $$ How would ...
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0answers
17 views

Error in cumulative distribution function due to error in mean

Suppose that $x_1,x_2,x_3$ are three random variables with mean $\mu_1=5, \mu_2=3, \mu_3=1$, respectively and each with variance $\sigma = 0.1$. The joint cumulative distribution function(cdf) of ...
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18 views

probability problem using Chebyshev's inequality

Suppose that a die has its "3" side changed to a "2". The problem is to first find a lower bound on the probability $P[3\leq X \leq 4]$ using Chebyshev's inequality. Then if we roll the die $n$ ...
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2answers
44 views

Expected value (and distribution) of sum of six balls labeled 1-49, no replacement.

The problem stems from the Spanish lottery, in which 6 balls are drawn from an urn with 49 balls, labeled 1-49, without replacement. My goal is to figure out the expected value of their sum, and if it ...
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1answer
46 views

Are $X$ and $X+Y$ independent, if $X$ and $Y$ are independent? [closed]

As asked in the title? Does the independence of two random variables $X$ and $Y$ imply the independence of $X$ and $X+Y$? If so, what's the easiest way to prove that?
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2answers
39 views

A problem in joint random variable (bivariate normal)

Suppose that $Y_{1}$ and $Y_{2}$ follow a bivariate normal distribution with parameters $\mu_{Y_{1}}=\mu_{Y_{2}}=0$, $\sigma^{2}_{Y_{1}}=1, \sigma^{2}_{Y_{2}}=2$, and $\rho=1/\sqrt{2}$. Find a linear ...
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1answer
33 views

Bias of $\sigma^2$ estimator

I need to find the bias of $\frac{\sum(x_{i}-\bar{x})^2}{n+1}$ for $\sigma^2$. To do so, one must take its expectation but add and minus $\mu$ from the summation part so we can bring $\sigma^2$ into ...
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2answers
42 views

joint distribution, discrete and continuous random variables

This may be trivial, but if X is a random variable uniformly distributed over $[0,1]$ and Y is a discrete random variable such that $\mathbb{P} (Y=y_1) = \lambda \in (0,1]$ and $\mathbb{P} (Y=y_2) = 1 ...
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1answer
19 views

Calculate the Probability of a Normally Distributed Random Sample

Please i would like to understand these problems about probability distributions, I can't find a right solution for this problem. I have a variable X which is the level of glucose in blood and is ...