# Tagged Questions

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

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### Continuity of the joint distribution function given continuity of marginals

Suppose $X$ and $Y$ are continuous random variables such that $F_X$ and $F_Y$ are the respective distribution functions. Suppose $F_X$ is continuous at $x_0$ and $F_Y$ is continuous at $y_0$. Then ...
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### Bounds-negative binomial distribution

Suppose $Y=\sum_{i=1}^{n} X_{i}$ where each $X_{i}$ is an independently and identically distributed geometric random variable with success parameter $p$, so that $Y$ has a negative binomial ...
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### Relationship between distributions of correlations $\rho(X^1,Y^1)$ and $\rho(X^2,Y^2)$ if $X^2=WX^1$, $Y^2=WY^1$ and $W$ is a known stochastic matrix?

I have been stacked for a while with the following problem: Consider two samples of iid observations $X^1=\{X_1^1,\dots,X_n^1\}$ and $Y_1=\{Y_1^1,\dots,Y_n^1\}$ where $X_i^1 \sim \mathcal{N}(0,1)$ and ...
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### Computer Component with Gamma Distribution? [closed]

I comes to a question of one old-exam as follows: if the life of one computer component (in year) has Gamma Distribution (if I translate correctly) with ...
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### Two company and probability example?

I ran into a problem that seems strange to me. Two companies A,B produce a device that with probability $0.05$ and $0.01$ are broken. if we buy two devices produced by one company with equal ...
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### Stochastic dominance of Binomial and Poission

In order to investigate the size of the cluster of a given vetex in a random graph I need to use a fact about stochastic dominance that I don't know how to prove. Namely, I am looking for a proof of ...
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### Is the support of the Gaussian finite or infinite?

Considering that as $x \to \pm \infty$ ; $e^{-\frac{x^2}{2}} \to 0$, is the support finite or infinite? A simple enough question, but enough to make me scratch my head. I feel that it's almost a ...
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### Probability density function for distance between two points.

Two points are chosen randomly inside a circle (and even on the circumference) with radius $r$ What is the probability density function of the distance between the points? I would be very grateful.
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### Differentiability of CDF at 0

This might seem to be a very trivial question but anyway here we go: I'm currently reading the paper "On the Value of a Random Minimum Spanning Tree Problem" by Frieze (1984) and I'm stuck on the ...
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### Finding density and distribution functions [closed]

I have been trying to understand probability by attempting past paper question and I have been stuck on this question all day and night. I am not quite sure how to go about finding the functions ...
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### A problem on probability concerning distributions of particles

I saw this problem in An Introduction to the Theory of Statistics by Mood, Graybill, and Boes (2nd ed.). I am quite intrigued by the problem. Here it is: Suppose that a particle is equally likely ...
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### Looking for a distribution to describe 2D “lines”

I have a 2D surface endowed with line segments, i.e. the function contains sparsely distributed segments in which there is high correlation between adjacent points in some direction, and the rest of ...
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### Binomial? probability of two producing defective articles machines.

Suppose that machine A produces (on a daily basis) twice the articles that machine B produces. However, 4/100 of the articles produced by machine A are defective while 2/100 of the articles produced ...
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### Could we define two random variables such that the product of them is Normal distribution(Gaussian)?

Could we find two random variables $X$ and $Y$ which $XY \sim N(\mu, \sigma^2)$? I found the ratio of two normal distributed random variables is distributed Cauchy distribution. However, on the ...
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### Constructing a piecewise probability density function

Here I'm trying to construct a probability density function in the form $$f(t) = \begin{cases} at, & t \in [0, 5) \\ b\sqrt{t}, & t \in [5, 20]\text{.} \end{cases}$$ Of course, \int\limits_{...
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### 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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### 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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### 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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### 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 $x \geq 1 \implies F(x) = 1$...
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### 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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### 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 ...
If $X$ and $Y$ are independent and exponentially distributed, which is the pdf of $Z$? Where $Z$ is given by $$Z = \frac{X}{1+Y}$$ I read answer to this post: $X,Y$ are ...
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 ...