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

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48 views

What is cosX-cosY when X and Y are uniform random variables

Let $X$ and $Y$ be iid uniform $$X,Y \sim U[-\pi,\pi]$$ Consider the following $$ U = cos (X)$$ $$V=cos(Y)$$ What is the distribution of $$W=U-V$$ I know that $$f_{U}(u) = \frac{1}{\pi\sqrt{1-u^2}...
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66 views

logic behind Negative binomial distribution

Is reaching 30th failure before 3rd success equivalent to reaching 0,1 or 2 successes before 30th failure? that is if Y counts number of successes before k-th failure and \begin{equation} \mathbb{P}(...
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1answer
502 views

Find the probability of at most two accidents per day in total on these highways

this is a question in my textbook that doesn't have a solution. Any help on an answer would be great. There are three highways in the county. Independently from one another, the number of daily ...
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1answer
22 views

Finding a good distribution

I'm given the following problem: There are a mouse and two chickens in a cage. The probability that each of the animals escape from the cage in the next hour is $0.3$. We wait an hour and observe what ...
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103 views

Two independent Poisson processes.

I am trying to prove the result that exactly $k$ occurrences of a Poisson process before the first occurrence of another independent Poisson process is a geometric random variable. \begin{align} &...
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63 views

$\mathbb{E}(\alpha^Y)$, where $Y$ is negative binomial

It is given that $\alpha>0$ and that \begin{equation} \mathbb{P}(Y=y)=\begin{pmatrix} y+k-1\\ y \end{pmatrix} (1-p)^kp^y \end{equation} are there any ideas how to calculate expected value of $\...
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1answer
55 views

How to calculate $P(X>a|X>a+b)$ and relate the meaning of a conditional equation to a real life situation

What I did: I found the Cumulative distribution, which is: $$P(X\leq x)=1-e^{-\frac{x}{\theta}}$$ Then I know that : $$P(X>a+b|X>a) \cdot P(X>b) = P(X>a|X>a+b) \cdot P(X>a+b)...
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295 views

Box Muller Transform - Proving that Z is Normal Distribution

I'm studying the Box Muller transform and I cannot see how Z0 and Z1 represent standard normal distributions. I've looked at the wikipedia page for the box-muller function but they don't seem to have ...
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1answer
69 views

Challenging Joint Probability Question

This is quite a challenging question I've been having trouble with lately and seems to be an atypical example of a joint distribution question - While I can derive the marginal density of $X$ simply ...
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1answer
17 views

Prevalence estimates based on randomized sample of clinical data

This is probably one of the more straight forward questions on here but here it is: I want to use a random number generator to sample X number of charts to look for the # occurrences of Y event. So ...
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1answer
133 views

CDF, PDF, and integral with respect to a function

In a certain textbook, I see the Cumulative Distribution Function (CDF) of a continuous random variable X defined as $$\int_{-\infty}^{x'} dp(x)$$ where p(x) is the Probability Density Function of X. ...
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1answer
26 views

An Airplane Probability to reach Oklahoma??

An airplane is built to be able to fly on one engine. If the plane' s two engines operate independently, and each has a $1\%$ chance of failing in any given four-hour flight, what is the chance the ...
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1answer
35 views

Moment-generating functions problem, total time before success at performing a task?

I'm facing moment-generating functions for the first time so I'm trying to get some practice and I'm stuck... Someone's trying to succeed at performing a given task. At each trial he either stops ...
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1answer
58 views

Challenging Marginal Probability Question

I've been stuck on this question for some time now - While I can derive the marginal density of X simply from the uniform distribution, I am at a loss for how the marginal density for Y and the joint ...
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1answer
81 views

First-order stochastic dominance and truncation

Suppose we have two distributions $F$ and $G$ over $\left[0,1\right]$. Suppose $F(x) \leq G(x)$ for all $x$, i.e. $F$ first-order stochastically dominates $G$. Is it true that $F(x|x\leq k) \leq G(x|x\...
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61 views

Finding the variance of speeds

This is a question from my Statistics textbook which I am currently stuck on. I have approached the question in a couple ways but each time I have been incorrect. A summary of the speeds, x ...
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1answer
41 views

compound poisson

The variable S has a compound Poisson claims distribution with the following: Individual claim amounts equal to $1$, $2$, or $3$. E(S) = $56$. Var(S) = $126$ $\lambda = 29$ Determine the expected ...
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3answers
1k views

Suppose 5 red and 7 green balls are in a bag. Three balls are removed without replacement.

Suppose $5$ red and $7$ green balls are in a bag. Three balls are removed without replacement. What is the probability that the second and third balls are both green? I'm having trouble figuring out ...
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1answer
74 views

Generating a rate equation from a paper

I'm going through equations in this paper Structure of Growing Networks with Preferential Linking, I was not able to understand how they derived equation $[3]$ by summing up equation $[2]$. eqn [2] $...
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1answer
32 views

generating a random probability mass function with uniform $p_i$'s

I would like to find the random probability mass function of a tuple $(p_1, p_2, ..., p_n)$ such that each variable is distributed such that $$p_i \sim U(0,1)$$ individually, but that every tuple ...
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2answers
280 views

Interesting Probability Question - Birthday Problem Variation

Suppose at a hipster eatery they make craft pickles. At this eatery they have $n$ pickle makers (picklers). Every day each pickler makes $10$ jars of pickles. Whenever any pickler has a birthday, ...
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207 views

The bug “probably” gets stuck!

Consider a regular tetrahedron with vertices $A,B,C,D$. A bug starts crawling from $A$. The bug moves from one vertex to the other along the edges continuously until it reaches $D$, where there is ...
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1answer
62 views

Expected Min/Max when picking numbers between 0 and 1

Suppose you pick two numbers between $0$ and $1$ with each number being picked at equal probability. What is the expected min/max of these two numbers? What if you picked ten numbers from $0$ to $1$? ...
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1answer
141 views

Assistance please on distribution problems

How many ordered quadruples (a,b,c,d) satisfy a+b+c+d=18, where a,b,c,d are odd positive integers? How many ordered quadruples (a,b,c,d) satisfy a+b+c+d=18, where a,b,c,d are integers such that |a|,|...
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63 views

CDF to Borel measure

If I have a right-continuous nondecreasing function $F:\mathbb{R} \to (0,1)$ that tends to $0$ and $1$ as $x$ tends to $-\infty$ and $\infty$ respectively, does $F$ necessarily induce a Borel measure $...
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1answer
107 views

Distribution of functions of uniform random variables

Given these two independent and uniform distributed random variables, $$X \sim U[-\pi,\pi]$$ and $$Y \sim U[-\pi,\pi]$$ What is the distribution of $$\sin(X)$$ and $$\sin (Y)$$ and the distribution ...
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1answer
25 views

Calculating the distribution, expected value and variance

I have no idea how to solve the following problem. Could someone give me some pointers on how the solve the following problem? Choose a random country. Taken $n$ persons from this country. ...
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1answer
98 views

What is the distribution of functions of Nakagami?

Assume we have $U_i$ Nakagami distributed with parameter $m$, for $i\in [1,n]$. What would the distribution of the following $$ \big|\sum_i {a_i U_i}\big|^2$$ where $a_i$ are non-negative constants. ...
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1answer
37 views

Almost a Frechet distribution but not quite yet

I have function as $$\frac{2}{\alpha}x^{\frac{2}{\alpha}-1}e ^{-x^{\frac{1}{\alpha}}}$$ This kind of reminds me of the Weibull and Frechet distribution but not quite because if it were I should be ...
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22 views

Random graphs: A random graph induces a distribution on each link. Do they uniquely determine the graph?

Let $G$ be a random graph with $n$ nodes, with the nodes numbered. $G$ induces a distribution in the set of all graphs of $n$ nodes and we can identify this distribution with $G$. Given the nodes $i,...
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2answers
2k views

Is the y axis on a PDF actually meaningless?

This idea popped in my head when I was reading this post on the normal distribution and the y-axis. My question is (and taking advantage of a nearby computer), a PDF inputs one value and returns ...
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1answer
357 views

What is the joint distribution of sample mean and sample variance of normal distribution?

${X_i} \sim N\left( {\mu ,{\sigma ^2}} \right)$, define $\overline X =\dfrac{1}{{n}} \sum\limits_{i = 1}^n {{X_i}} $, ${S^2} = \dfrac{1}{{n - 1}}\sum\limits_{n = 1}^n {{{\left( {{X_i} - \overline X} \...
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1answer
61 views

Meaningful statistic measure of data pairs

I have a dilemma. I have pairwise data, (a,b), that represents some form of speed, whether it's miles/hour or megabits/second. Let's say that we have the following set of data from measuring the "...
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2answers
256 views

Probability (X >Y) when X and Y have the same distribution?

This is a problem from HW4 Joe Blitzstein's Harvard Stat 110 course. Let X be a random day of the week, coded so that Monday is 1, Tuesday is 2, etc. (so X takes values 1, 2, . . . , 7, with equal ...
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1answer
51 views

How to find the $E(N)$ using $E(M)$ where the $M$ and $N$ follow slightly different scenarios

An author sends his first manuscript to a large number of publishers, $C, D, E, ...$ , in turn, only approaching each one, after the first, if the one before has refused it. There is a constant ...
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1answer
207 views

Equal in distribution but unequal almost everywhere?

If this question has been asked, I apologize but I could not find it. I was wondering if it was possible construct $X$, $Y$ two iid rv's such that they equal in distribution, i.e. $P_X(B) = P(X^{-1}(...
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35 views

Computing a Finite Expectation

Assume $1\leq\ k<m<n$ are positive integers and $X_1,X_2,...X_n$ are i.i.d. Geometric($p$) random variables. For all $j\geq\ k$ define $I_j=[(i_1,i_2,...,i_k):1\leq\ i_1<i_2<...<i_k\leq\...
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70 views

Sum of Lomax random variables

Suppose $X_1,X_2,\cdots X_n$ are $n$ i.i.d Lomax random variables with pdf $f(x)=\frac{m}{(1+x)^{m+1}},x\geq 0,m\in \mathbb N$. I need to determine the pdf (or cdf) of the sum $S_n=\sum_{i=1}^{n}X_i$. ...
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43 views

For fun, how many paths are there in a flow matrix?

I just got done doing the whole flow-matrix percolation exercise in programming. That is a situation where you have, for simplicity, an $n\times n$ grid of 0s and 1s. 0s represent blocked sites and ...
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1answer
236 views

Probability from multiple trials

This questions is from a practice mid-term that I don't have a solution to. A monkey in a research lab is given 6 tiles with the letters AAABNN. On each trial the monkey randomly arranges the ...
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37 views

Why the doubly non-central F distribution does not have a mean or variance if the denominator degree of freedom is less than or equal 2 ??

Normally the doubly non-central F distribution is generated by the division of two non-central chi squared Random Variables,, so what is the the problem of using any famous formula to get the mean of ...
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1answer
25 views

What is the probability of success?

If I have 12 Possible questions, of which 5 are asked and I only need to answer 2 of them, what is the probability of my success (i.e., I am able to answer 2 of the 5 asked questions) if I learn 2 of ...
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3answers
60 views

Symmetric Distribution of Random Variable

Prove: Let $X$ and $Y$ be random variables with the same distribution. If $X$ and $Y$ take only two values​​, then $X - Y$ are symmetrically distributed around zero. Note: 1 - You can use ...
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2answers
74 views

Conditional Probability - chance for an event to happen

I am learning probabilities at the moment and I have come across this problem: A person takes four tests in succession. The probability of his passing the first test is p, that of his passing each ...
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2answers
48 views

Probability CDF question on highest number of marbles pulled out

I'm kinda stuck on this problem. Here goes: An urn contains n marbles, numbered 1, 2, . . . , n. Suppose k < n marbles are drawn from it at random without replacement. Let X denote the highest ...
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2answers
404 views

A random variable $X$ uniformly distributed over the interval $[0, 2\pi]$

A random variable $X$ distributed over the interval $[0, 2\pi]$ a) the pdf of $X$ b) the cdf of $X$ c) $P(\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$ d) $P(-\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$ ...
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3answers
94 views

Find the pdf of T = X + Y

Let (X,Y) be a random point chosen uniformly on region R = {(x,y) : |x| + |y| <= 1}. I need to find the pdf of T = X + Y. I know the joint density is just equal to 1/(area) = fxy(x,y) = 1/2 for |x|...
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83 views

Probability Distribution for a Weird Card Game

I promise this is not for a homework problem, even though this sounds like only something a professor would dream up. Here is the game: I begin with a deck of 13 cards: 1 through 10, Jack, Queen, and ...
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22 views

How is this Variance found in this old question?

On this question: Probability: Normal Distribution they find these values: $\hat\mu = .05(150) = 7.5\space,\hat\sigma = \sqrt{150(.05)(.95)} = 2.67$ I see how they got $\mu$, but how did they get $\...
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110 views

Right continuity of right inverse of right continous map

I am stuck on the following proof that I found in Dellacherie-Meyer's book "Probabilities and potential B", p. 119 (increasing processes and projectors). Given a map $a$ on $[0,\infty [$ which is ...