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

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1answer
24 views

Find limit distribution i.i.d $\xi_1,\xi_2\dots$ uniformly on $[0,1]$

Let $\xi_1,\xi_2\dots$ independent and identically distributed uniformly on $[0,1]$ and $\zeta_n = \min(\xi_1,\dots,\xi_n)$. Find limit distribution $n^{\gamma}\zeta_n$, $\gamma\in R$. My try. ...
0
votes
1answer
11 views

Distribution of exponential(X/c)

Suppose $X \sim Exponential(\lambda)$. That is, the PDF for $X$ is $f_X(x)=\lambda \cdot e^{-\lambda x}$, $x\ge 0$, and the CDF of $X$ is $F_X (x)=\int_{-\infty}^x f_X(x)=1-e^{-\lambda x}$, $x\ge ...
-2
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0answers
23 views

The mean deviation from mean in a normal distribution is equal to $4\sigma/5$ [closed]

Show that the mean deviation from mean in a normal distribution is equal to $4\sigma/5$. Progress. I have tried going by the usual definitions of deviation and mean deviation but am stuck. Tried ...
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1answer
24 views

If you have 50 envelopes and only 3 envelopes contain a symbol what is the probability of picking all 3? [closed]

If you have 50 envelopes and only 3 envelopes contain a symbol. the person picks only 3 envelopes out of the 50. What is the probability that they will pick 1 symbol? Two symbols? All 3 symbols?
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0answers
11 views

Given a sample determine using Chi-squared test whether these values fit in an EXPONENTIAL distribution

Here I've got such a problem. I was given $n = 20$ values for time of good functioning of a robot between two consecutive defects. 1200, 1432, 1502, 1100, 3286, 4235, 1149, 5236, 2234, ...
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0answers
6 views

Limit distribution of absolute value maximum of stationary non-differentiable Gaussian process

Consider a real-valued stationary Gaussian Process $\{ X(t) \colon t \geq 0 \}$ with zero mean and unit variance and covariance function $r$ satisfying $r(t) = 1 - C|t|^{\alpha} + o(|t|^{\alpha}), ...
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0answers
9 views

Density of the Absorbed Process

The curiosity arose while reading the Ch.18 of Arbitrage Theory in Continuous Time 3/ed, dedicated to pricing Barrier Options. Definition 18.1 For any $y\in R$, the hitting time of y, $\tau(X,y)$, ...
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1answer
11 views

Relation between joint probability and marginals for two dependent random variables?

Consider two continuous real valued random variables $X$ and $Y$. Let $f(X,Y)$ be their joint probability distribution and $f_X (X),f_Y(Y)$ their marginals. Suppose that $X$ and $Y$ are dependent. Is ...
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0answers
19 views

How to compute the average power of an ergodic process?

Rxx(0)=3 is the average power and if i take limit as t goes to infinity i will get the (E[x])^2 to get variance you subtract 3-2 = 1 is this correct ? and can someone tell the difference ...
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0answers
14 views

How to compute a probability expression (for a transition matrix of a Markov Decision Process)? (part 2)

I am creating a transition matrix (for a Markov Decision Process) and I am computing it using a Matlab script, which I am currently writing. My probability expression (for certain cases) looks like ...
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0answers
19 views

How to compute a probability expression (for a transition matrix of a Markov Decision Process)? (part 1)

I am quite new in the world of statistics, hence I am quite unsure when working with probabilities. I am creating a transition matrix (for a Markov Decision Process) and I am computing it using a ...
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0answers
9 views

Chernoff type bounds for negative binomial distribution

If I recall correctly I remember reading that we cannot get Chernoff type results for the negative binomial distribution because of something regarding lebesque measure. I don't quite know all the ...
1
vote
1answer
28 views

Expectancy of a joint density

A machine consists of two components, whose life times have the joint density function $ f(x,y)= \begin{cases} 1/50, & \text{for }x>0,y>0,x+y<10 \\ 0, & \text{otherwise} \end{cases} ...
0
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0answers
9 views

Decay time distribution with uniformly distributed source

Consider a kind of particle (source) that can decay into some other particle (product) with decay constant $\lambda$, i.e. the p.d.f is $f(t)=\lambda e^{-\lambda t}$, and the source is uniformly ...
1
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0answers
25 views

Survival probability (1D Brownian Particle)

Here is an interesting article from Wikipedia: First-hitting-time model I am particularly interested in how the following density is derived: $$p\left(x,t;x_0,x_c\right)=\frac{1}{\sqrt{4 \pi D ...
3
votes
2answers
31 views

Notation $E[t^X]$ where $X$ is a random variable

I have a quick question which occured in the context of probability-generating functions but maybe the issue is more basic. For a random variable $X$, the probability-generating function is given as ...
1
vote
2answers
26 views

Probability distribution of number of columns that has two even numbers in a chart

We distribute numbers $\{1,2,...,10\}$ in random to the following chart: Let $X$ be the number of columns that has two even numbers. What is the distribution of $X$? My attempt: ...
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0answers
7 views

specific examples of random variables satisfying a given condition.

Theorems such as the central limit theorem only says random variables satisfying certain conditions have some properties. Now, what I am curious about is the existence of such random variables. For ...
-1
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1answer
29 views

Birth and Death process, CTMC, how is the solution here derived? [closed]

My question is about how the solution is reached, as I am completely lost on how. Any thoughts? Consider a birth and death process with birth rates $λ_i = (i+1)λ \;\;, \;\; i≥0$, and death rates ...
1
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0answers
20 views

How to distribute two independent rows of bits

Consider two independent rows of 100 bits. The bits are mutually independent and have an equal chance to be 0 or 1. The first row is being read and during that process there is a chance $\epsilon$ ...
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votes
3answers
45 views

Finding Variance and Expectation of Boolean Variable

Below is the joint distribution of Boolean random variables X1, X2 and X3. How do I find variance and expectation of X2? I understand that variance is "average of squares of difference from mean ...
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0answers
20 views

Is there a Continuous Multinomial Distribution??

In Multinomial Distribution, we have \begin{align} f(x_1,\ldots,x_k;n,p_1,\ldots,p_k) & {} = \Pr(X_1 = x_1\mbox{ and }\dots\mbox{ and }X_k = x_k) \\ \\ & {} = \begin{cases} { \displaystyle ...
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0answers
11 views

Convergence in Distribution for two Dirichlet distributions

I'm working on a problem and I wanted to get some hints on how to solve it. To me, it seems like showing convergence to distribution but since it's been a while that I've not worked on these types of ...
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0answers
19 views

Continuous moment generating function but discrete distribution.

I have the following exercise; Suppose that X is a random variable for which the m.g.f. is as follows; $$\psi(t)=\dfrac{e^t}{5}+\dfrac{2e^{4t}}{5}+\dfrac{2e^{8t}}{5}$$ for all $t \in \mathbb{R}.$ ...
1
vote
1answer
26 views

Distribution of product of independent Gaussian random variables

Let $X,Y$ be i.i.d. $\sim N(0,1)$. Then $$\frac{XY}{\sqrt{X^2+Y^2}} \sim N(0,1/4).$$ How can I prove this? I've tried applying the transformation formula, but it hasn't worked out thus far.
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0answers
16 views

Covariance from Gaussian Mixture Model versus direct calculation

I have a random variable $T$ which can be expressed as follows: $T=c_1P\cdot f+c_2Q$, here $\cdot$ denotes point-wise multiplication. $c_1,c_2$ are scalars and treated as constants. Distribution of ...
-1
votes
0answers
28 views

Probability of getting 3 balls in 10 rooms with 9 people [duplicate]

There is a group of 9 people who visit 10 different rooms together. Each room has 3 balls, and each person has an equal probability of getting a ball. What is the probability that, after visiting all ...
0
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0answers
19 views

Mixture of Dirichlet Distributions

I'm working on a problem for Dirichlet distributions and I appreciate if you can give me some hints. Consider two random vectors of size K that are distributed as Dirichlet: $$\vec{Y_1} \sim ...
2
votes
1answer
26 views

When Benford´s Law doesn´t apply - p(d) = (10.5 - d)/(49.5) instead

When Benford´s Law doesn´t apply I would like someone validate (or not) the formula (A) Suppose I pick some book and open it at random. What the probability for first digit page being 1 ? Related ...
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1answer
23 views

Poisson distribution help?

The number of car accidents occurring per day on a highway follows a Poisson distribution with mean 1.5. a) Given that at least one accident occurs on another day, find the probability that more than ...
0
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1answer
18 views

PART TWO: Poisson counting process, probability system errors divided in time periods at a certain rate

I've been trying to apply the same knowledge from a previous post, but perhaps my reasoning is wrong. "Errors in a computer surfaces according to a Poisson process with rate 0.4 per day. If there has ...
0
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1answer
25 views

Interpreting a condition about CDF

Let F(X) be a strictly increasing CDF which admits a positive density f(x). Is the condition x/F(x) being non-increasing (aka, weakly decreasing) equivalent to saying that F(x) is convex? If not, what ...
1
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0answers
41 views

Finding probabilities of a continuous random variable

I have the following continuous random variable density function: $$ f(x) = \begin{cases} \frac14 & if\,0\le x<1 \\ \frac12 & if\,1\le x<2 \\ a & if\,2\le x<4 \\ 0 & ...
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0answers
3 views

Difficulty to be in first percentile as sample increases.

I was wondering: How can we explain that it is harder to be in the best percentiles as the sample grows ? This question comes from a game called League of Legends, where they introduced a system ...
2
votes
1answer
49 views

Calculate the probability select $k$ blue balls in box

I have a box that contains 10 balls( 2 red balls and 8 blue balls). Probability select each ball is an uniform distribution. An event is defined that selects k balls $(0<k\le 10)$ from the box and ...
1
vote
1answer
49 views

Understanding of the probability using poisson

I have a question about my statistics homework. The question is as follows: At an army base there are X number of soldiers hit by a car. The poisson distribution expaction of this is μ=2. The ...
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0answers
16 views

Find the PDF of $Y = a*X - b*X^3$ given that $X$ is a uniform random variable on $[0,1]$

Assume that $X$ is a uniform RV on $[0,1]$ and that $a$ and $b$ are both positive. Can also assume that $Y$ is monotonically increasing over its range. I'm trying to find the PDF of $Y$ and am ...
0
votes
1answer
28 views

Poisson counting process, probability system errors divided in time periods at a certain rate

I had some help on a previous post where I learned that The distribution of $X(t)-X(s)$, for $s<t$, is Poisson with rate $\lambda(t-s)$. That is, $$ \mathbb P(X(t)-X(s)=k) = > ...
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0answers
20 views

How to show analytically the pdf for the minimum of random variables

If $Y_1,Y_2,\ldots,Y_n$ are $i.i.d$ and each $Y_i$ is Generalized Gamma $GG(kn,\lambda)$ distributed. Assuming, the form of $Y_i = Z_i^m$ and each $Z_i^m$ is Gamma distributed, then what will be ...
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votes
1answer
19 views

Poisson counting process, probability of arrival of x customers at a certain rate [closed]

I do apologize on beforehand because this post will most definitely be downvoted due to (choose random reason) as I have no clue on how to approach these type of questions. I have been reading my ...
0
votes
1answer
21 views

How to compute selection probability of balls in a range

I have a question about probability that need your help. I assume that I have balls that are numbered from 1 to 100. The probability selection each ball is followed uniform distribution. I divide ...
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0answers
24 views

A proof involving conditional distributions, Poisson, binomial

Let $X$ be a non-negative integer valued random variable. Let $Y$ be the number of successes in $X$ binomial trials. Prove that, if the distribution of $Y$ and $X\mid (Y=X)$ are identical, then $X$ ...
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0answers
8 views

distribution of non-central chi random sample

Suppose that $X_1,X_2, \ldots, X_n$ is a random sample from a non-central chi distribution with $1$ degree of freedom. What is the distribution of the sample variance of $X_1,X_2, \ldots, X_n$?
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0answers
12 views

Working out closed form of shifted poisson distribution

In the article "Bayesian variable selection for Poisson regression with underreported responses" the author defines $t_i^0$ as the number of actual occurences in a study in the $i$th covariate ...
1
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1answer
19 views

Two stochastic variables

Say $Y$ is the nummer of accidents by a car, Poisson distributed ($\lambda$). People hit by the car have a probability $p=\frac{1}{2}$ to survive. Let $Z$ be the nummer of people killed by a car ...
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0answers
8 views

Probability density function with these properties

Consider a family of probability density functions $f$ with parameters $A, B, \mu, \sigma^2$, satisfying the following properties: First, the parameters $A$, $B$ should satisfy $A < B$ and ...
0
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0answers
11 views

Compute the expectation of a function of a random vector not knowing the whole distribution

Imagine I have three random variables $X,Y$ and $Z$. I know that $X\sim Y$ which does not imply that $(X,Z)\sim(Y,Z)$ I know the distribution of $(Y,Z)$. So, in a summary: I know the distribution of ...
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0answers
9 views

Finding the correct distribution

Consider the independent stochasts $X$ and $Y$ that take the values $\pm 1$ with an equal chance (i.e. $1/2$). I came to the following distribution: $f_{X}(x)=(1/2)^{x}*(1/2)^{1-x}=1/2, x=\pm 1$ and ...
1
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1answer
15 views

When two random variables that have the same law… Can they be happily exchanges?

Imagine, $X$ and $Y$ are two random variables which have the same law, which we denote by $X\sim Y$. We have then a third random variable $Z$. Can we say that $$(X,Z)\sim (Y,Z)?$$ In what cases is ...
0
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0answers
24 views

To prove that induced probability measure indeed defines a probability measure

Given two measurable spaces $(Ω_1, B_1)$ and $(Ω_2, B_2)$, a measurable function T : $Ω_1 → Ω_2$ and P is a probability measure on $(Ω_1, B_1)$. $B_1$ & $B_2$ are respective sigma algebras. The ...