Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

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11 views

Estimating probability density function of big amount of data coming from Monte Carlo simulation

I am trying to estimate Probability Density Function (PDF) of a big amount of data ($1e^6$ , $1e^7$, and higher) coming from Mote Carlo (MC) simulation. My objective is to estimate the PDF (e.g. with ...
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0answers
21 views

Is it possible to talk about the distance of two random variables?

Suppose that $X, X^\prime$ are two random variables with the same distribution $\mathcal{N}(\mu, \sigma^2)$. Is there anything that we can say about the "distance" between the two random variables? ...
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6 views

Relation between the probability and timing of occurrence of an independent and identically distributed (IID) event.

If the success probability of an IID event is $p$, what can we say about the occurrence timings of the event? A paper had the following lines: the event that a link is successfully activated in ...
2
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2answers
25 views

Covariance of continuous functions, uniform and normal distribution

For X~Uniform(1, 9.9) and Y|X = x~Normal(1.4, x^2) What is Cov(X, Y) equal to? What I tried was: E[XY] - E[X]E[Y] Where E[X] = 5.45 and E[Y] = 1.4 But for E[XY] I'm a bit clueless. I've considered:...
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1answer
17 views

Independence of two variables given a finite covariance with a third variable

Suppose that we have three continuous variables A, B, and C, whose joint probability distribution is P(A,B,C) = P. We are given that the two covariances $\langle AB \rangle - \langle A \rangle \...
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1answer
13 views

Simulated & Theoretical Values

How do I justify the difference in the values obtained via theoretical calculations and simulated calculations with random numbers? Is it always true that theoretical calculations yield higher ...
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2answers
33 views

Expected values of dependent Gaussian variables

Let $(X_1,X_2,X_3)$ be a set of three zero-mean Gaussian random variables with a covariance matrix of $$ C=\sigma^2 \begin{bmatrix} 1 & \rho & \rho \\ \rho & 1 & \rho \\ \rho & \...
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1answer
11 views

What is the joint probability density function for two continuous distributions of iid random variables of different type?

Let $X$,$Y$ be independent and identically distributed continuos random variables. $X$ is distributed according to a Gaussian distribution with pdf ${f_g}_X(x;\mu,\sigma)$, while $Y$ according to a ...
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0answers
16 views

Distribution formed by functions of elements of a multivariate normal distribution

If you have a multivariate normal (MVN) distribution of format $$ \begin{pmatrix} a \\ b \\ c \end{pmatrix} \sim MVN \left( \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \begin{...
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1answer
35 views

Need help plotting the following function: [on hold]

X-axis: $ a*(1-b)$ Y-axis: $ b*(1-a)$ where both a and b are probabilistic values that lie between 0 and 1.
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1answer
20 views

What is the probability that the second customer arrives at least $k$ times faster than the previous one?

The exact question is not quite the same (it was hard to put in a concise title) but has the same principle: The time between customers has an exponential distribution. What is the probability that $...
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1answer
29 views

Mean and variance of pdf which is close to standard normal distribution

I'm trying to calculate mean and variance of a distribution with the following PDF $$p(t)=\frac{e^-\frac{(t-a)^2}{2\sigma^2}}{\sigma\sqrt{2\pi}}+m\frac{e^-\frac{(t-a)^2}{2\sigma^2}}{\sigma\sqrt{2\pi}}(...
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2answers
48 views

Find the unique distribution of a random variable knowing the moments of the random variable

This problem comes from Allan Gut's 'An Intermediate Course in Probability', but I cannot solve the problem. The random variable $X$ has the property that $$EX^{n}=\frac{2^{n}}{n+1}, \quad n = 1,2, .....
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1answer
32 views

Given X~Exp what is the distribution of Y = X - floor(X)

This is from an exercise which my friend and my can't fully wrap our heads around. Given $X\sim Exp(\lambda)$ what is the distribution of $Y = X - \texttt{floor(}{X}\texttt{)}$ Approach: Compute pdf ...
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1answer
18 views

Two sided normal p-value question

From Statistical Inference by Casella and Berger: Let $X_1 , \dots, X_n$ be a random sample from a $n(\mu, \sigma^2)$ population. Consider testing $:H_0: \mu = \mu_0$ verses $H_1 : \mu \neq \mu_0$....
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1answer
27 views

Show that f is density of bivariate normal distribution

Let $\mathbb{X} := (X_1,X_2)$ be a random variable with given density function $$f_{\mathbb{X}}(x_1,x_2)= \frac{\sqrt{2}}{\pi}\exp\left(-\frac{3}{2}x_1^2-x_1x_2-\frac{3}{2}x_2^2\right),\space \text{...
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0answers
31 views

An important proposition of the Kolmogorov–Smirnov test

How do I prove the following theorem: The distribution of the Kolmogorov-Smirnov test statistic $D_n$ is, under the Null hypothesis $H_0: F = F_0$, equal for all continuous distribution $F_0$. ...
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1answer
19 views

how to explain $X$ and $Z$ Not relevant, how to prove it?

Let the random variables $X$ and $Y$ be independent,$X$ follows the exponential distribution with parameter $1$, and the probability distribution of $Y$ is $$\mathcal{P}\{Y=-1\}=p,\;\mathcal{P}\{Y=1\}...
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1answer
15 views

Expected Number of Events over Multiple Poisson Intervals

Let's say I find that in an interval of 5 ns, with event rate 50000/s, that the probability of an event occurring twice in that interval is $$P_2 = {e^{-\lambda} \lambda^{2} \over 2!} = 3.124 \cdot 10^...
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1answer
46 views

Solving the quadric equation: $\frac{-U\pm \sqrt{U^2-4V}}{2}$ where $U,V$~$\mathbf{U}(-1,1)$ independently.

I'm given two uniform random variables $V,U \sim\mathbf{U}(-1,1)$. I also get the function: $$h(s)=s^2+Us+V$$ I'm interested to answer queations like for which values $h(s)$ has only: zero /one/ two ...
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0answers
11 views

Raikov's theorem and shifted Poisson distributions

I am having trouble understanding the original statement of Raikov's theorem and its implementation in terms of characteristic functions. Raikov's theorem states that a shifted Poisson distributed ...
2
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1answer
21 views

Dominated convergence theorem applied to the convergence of measures

In this answer there was used the dominated convergence theorem. However, I don't see how it works here. It was said that it can be used with respect to the counting measure because the function $f$ ...
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1answer
29 views

Identify the Couples in a group of people

I'm reviewing for my stat midterm and I happen to block at this question. I tried to choose out of the women 5 and out of the men 5 multiply both and divide by the total (17 choose 10 since 5 men and ...
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0answers
22 views

The nature of total service time (response time) in a queue

Considering a general queue with no specific characteristic (like a G/G/k/k). Is the nature of response time $T$, consisting of both service time and waiting time, identical to service time? For ...
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2answers
52 views

closed form for $\sum_{k=0}^{v-1}\binom{n-v-1}{k}(\frac{p}{q})^k$

In an exercise involving probabilty, in which $p=1-q$, and $v$ is a given positive integer I try to show that $\displaystyle \sum_{n=0}^{+\infty} t^v p^v (1-p)^{n-v} \sum_{k=0}^{v-1}\binom{n-v-1}{k}\...
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0answers
32 views

Is there any way to make a random sample from the same distribution, i.i.d when it is positively correlated? [on hold]

Is there any way to make a random sample from the same distribution i.i.d when one knows, is given that the subjects have been self selected and and as such are positevely correlated but knows linear ...
0
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1answer
25 views

Indicator of two exponential random variables

I am working on the following problem from a statistics qualifying exam. I have attempted progress at the 3 parts but do not feel certain, or satisfied, with my work. I am not sure if $\textit{I}$ am ...
2
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1answer
46 views

Distribution such that Expected Maximum is close to Expected Sum

For a distribution $\mathcal{D}$ on the nonnegative reals with expectation equal to 1, let $X_1$, $X_2$, $\ldots$, $X_n$ be $n$ independent samples. If we want to bound $\mathbb{E}[\max_i X_i]$, a ...
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1answer
48 views

Drawing balls from an urn

I'm preparing for my midterm and I'm having trouble connecting all the dots in the following problem: From an urn containing 2 yellow balls, 3 blue balls and 22 red balls, one ball is drawn, ...
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0answers
29 views

Brownian motion and Beta distribution

I am interested in the distribution of the time that the standard Brownian $W_t$ motion on $[0,1]$ satisfies the following inequality: $$W_t \ge stW(1)$$ For different values of $s$. I conjecture that ...
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1answer
42 views

Probability generating function of $X\sim \text{Poisson}(\lambda)$ when $\lambda\sim U(0,2)$

The probability generating function (pgf) of $X\sim \text{Poisson}(\lambda)$ is $$G_x(t) = e^{-\lambda(1-t)}.$$ Find pgf of $X$ if $\lambda\sim \text{Unif}(0,2).$ Then find $\mathbb P(X=2).$ My ...
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1answer
46 views

Distribution function of $\sin(\pi\theta)$ when $\theta\sim U(-1,1)$

If $\theta\sim Unif[-1,1]$, then what is the CDF of $U=\sin(\pi\theta)$? Now, its easy to see that $$P_{U}(t) = P\left(\theta \leq\frac{\sin^{-1}(t)}{\pi}\right)$$ somehow the answer is equal to : ...
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0answers
6 views

Asymptotic behavior of the maximum of independent but non-identically distributed Gaussian random variables

Is there any result on the asymptotic behavior of the maximum of independent but non-identically distributed Gaussian random variables? Something similar to the result by Gnedenko, (1947) that for a ...
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1answer
34 views

Relation between $\sup_{0\leq s\leq t}|W_s|$ and $\sup_{0\leq s\leq t}W_s$ for Brownian motion $W$

Let $W$ be a Brownian motion. In a calculation, I have to compute $$\mathbb P \left(\sup_{0\leq s\leq t}|W_s|>a\right).$$ My idea would be to use the reflexion principle that says that $$\mathbb P\...
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7 views

Distribution Pruning

I'm searching for some justification, what distribution should be chosen for elements of a certain size distribution $n> 1$, when they arise by choosing, for example, $k$ elements with the highest ...
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1answer
14 views

Expected value of zero-inflated model

Suppose I have some quantity $L$ which can be $0$ with probability $(1 - \theta)$ or can be distributed according to a lognormal $\mathcal{LN}\left(\mu, \sigma^2\right)$ with probability $\theta$. ...
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0answers
17 views

probability distribution for the following sequence of numbers [on hold]

I have dag (directed acyclic graph) with one root. Each node has some directed links to a set of other nodes . each edge is labeled with a number which represents the transition probability from the ...
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1answer
15 views

Equality in distribution of a sum

When I have 4 random variables, $A,B,C,D$ and know that $A+B \stackrel{d}{=} C + D$ and $A \stackrel{d}{=} C$, does this imply $B \stackrel{d}{=} D$? Going through the definition $X \stackrel{d}{=} Y$...
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0answers
12 views

Probability Beta distribution B1 less than Beta distribution B2

I am looking for the probability that a variable drawn from a Beta distribution $B_1$ with parameters $\alpha_1$, $\beta_1$ is less than a variable drawn from an independent other Beta distribution $...
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2answers
22 views

Negative log of unit scale gamma distribution converges to standard exponential

Let $Y \sim \text{Gamma}(\alpha,1)$, with pdf $f(y) = y^{\alpha-1}e^{-y}/\Gamma(\alpha)$, where $\alpha \in (0,1)$. Let $Z = -\alpha \log Y$. Prove that $Z$ converges in distribution to a standard ...
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1answer
53 views

Why are theorems stated for $u(X)$ instead of just $X$?

In Introduction to Mathematical Statistics by Hogg and Craig, they state the generalization of Chebyshev's Inequality as: Let $u(X)$ be a nonnegative function of the random variable $X$. If $E[u(X)]$...
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1answer
24 views

Probability distribution of people

There is have a group of 50 people where 30 are men and 20 are women and they are being separated into two equal classes of 25 people, what is the probability that any of the two classes will have 15 ...
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14 views

Showing $E(X^k)=\frac{r}{p}E(Y-1)^{k-1}$ where $X,Y$ are negative binomial

Let $X\sim\text{NB}(r, p)$. (a) Show that $E(X^k)=\frac{r}{p}E(Y-1)^{k-1}$ where $Y\sim\text{NB}(r+1, p)$. (b) Use this result to find $E(X^2)$. I am having trouble specifically with part ...
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0answers
19 views

“semi-circular” / “semi-directional” statistics

Directional (or circular) statistics deals with random variables "distributed over a bounded region". A classical example is e.g. the angle representing wind direction, which is distributed over 360°. ...
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0answers
27 views

What curve is this and where is it used?

While playing around with the normal distribution, or more precisely, the probability density function: $$\displaystyle f(x | \mu,\sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{(x - \mu)^2}{2 ...
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0answers
40 views

Deriving a Wishart distribution

I have a real $n \times n$ matrix $\textbf{A}$ with columns $\textbf{a}_1, ... \textbf{a}_n, \textbf{a}_j \sim N(\textbf{0},\textbf{S}_j)$. At some point I arrive at a p.d.f. kernel which is ...
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1answer
23 views

Probability law under change of measure

I have a question regarding the law under a change of measure. The radon nikodym theorem says, that $E_Q(X) = E_P(ZX)$ where $Z = dQ/dP$ the change of measure. I am not interested in the expectation,...
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1answer
23 views

Extinction probability of modificated branching process

From An Intermediate Course in Probability by Allan Gut: Consider the following modification of a branching process: A mature >individual produces Children according to the generateing function g(...
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1answer
27 views

Statistical Significance for a single patient

I have the mean values for a blood test from only 1 patient before and after treatment. My supervisor asked me to find statistical significance. I have the mean value of 4 parameters. Would it make ...
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19 views

Statistics and Probability Expected Value Questions

Compute the frst two moments (that is EY and EY^2) of Y = 2^-X if X = b(n, p) (binominal distribudion). Random variable X has the distribution P(X = k) = c*3^-k(keN). Determine the value of c. Does ...