Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
4 views

Variance of the range for the exponential distribution

If $n$ random variables are independently distributed with an exponential distribution $f_X(x) =\lambda e^{-\lambda x} (x\geq 0)$, the range $R_n = \max(X_1,\cdots,X_n) - \min(X_1,\cdots,X_n)$ has the ...
0
votes
0answers
4 views

Determining whether random sample variables of specific distributions are independent

Suppose we are given a random sample $X_1, X_2 , \dots , X_n$ with a pmf that corresponds to a known distribution which involves independent trials (such as the Binomial, or the Geometric, etc ) and ...
1
vote
1answer
21 views

let $X$,$Y$ be independent Poisson distributed random variables with parameter $\alpha$ and $\beta$ respectively. $E(XY)$?

$$E(XY) = \sum_{x,y} xy f(x,y),$$ but I don´t have the $f(x,y)$. $X+Y$ would be Poisson distributed with parameter $\alpha + \beta$, but what about $XY$? Not sure what else to do. Thanks in advance.
1
vote
0answers
9 views

Token Bucket Algorithm

Can the token bucket algorithm solve the below problem? Say I'm selling a product online and announce this to 1 million people. Assume incoming traffic will look similar to a Poisson distribution. ...
-1
votes
0answers
9 views

How to calculate the expected value of P{X<Y}? [on hold]

For these two random variables of X and Y, my question is how can I calculate the $E\{P(X<Y)\}$? Thanks a lot in advance.
-4
votes
0answers
21 views

PROBABILITY&STATSTCS [on hold]

In order to test the durability of as new paint, a highway department has test strips painted across heavily traveled roads in eight different locations. If, on the average, the test strips disappear ...
1
vote
1answer
20 views

What is the PMF of the product of two discrete random varibales? [on hold]

Let $X$ , $Y$ be two discrete random variables. What is the probability mass function of $Z=X Y$?
0
votes
1answer
15 views

Complicated Demonstration - Violation of the theorem that converges in probability and not in distribution

I was thinking that if a sequence of random variables $Y_n$ with c.d.f. $H_n$ which converges to $c$ in probability, such that $H_n(c)$ does not converge to $H(c)=1$. How could I make an example ...
0
votes
0answers
12 views

Rao Blackwell and sufficient statistics

Suppose that X1, . . . , Xn are independent identically distributed random variables with a B(m, θ) distribution where m is a known positive integer and θ is unknown. I have shown that θ* = X1/m is ...
2
votes
1answer
16 views

Rao-Blackwell and Cramer-Rao LB comparison

Let $X_1, X_2, \dots, X_n$ be a random sample following the Geometric distribution. $$ \prod\limits_{i=1}^{n} f(x_i|p) = (1-p)^{\sum\limits_{i=1}^n x_i-n}p^n $$ Since the pmf of the Geometric ...
0
votes
1answer
29 views

how to derive this property of exponential distribution?

Let $X$ be a real random variable with exponential distribution on a measure space with probability measure $\Bbb P$. Let $\Bbb E$ be the expected value. Then \begin{equation}\Bbb P(X>s+t|X>s)=\...
-2
votes
0answers
22 views

According to the uniform distribution U(0,θ), how to obtain the suitably normalized limit distribution for ((n+1)/n) X(n) )

I was thinking of doing it by means of the maximum likelihood estimator but how can I apply it to the case in which I get (n + 1) / n?
0
votes
0answers
16 views

Is the zero truncated Poisson Distribution part of the Exponential Family?

This is the density of a truncated Poisson: $$P(X = x \mid X > 0) = \frac{\lambda ^ x e^{- \lambda} }{x ! \left ( 1 - e^{- \lambda} \right )}$$ To show that it's member of the Exponential ...
0
votes
0answers
23 views

Does $X_1,…,X_n$ being a random sample from $N(\mu,\sigma^2)$ $\implies \frac{\overline{X}-\mu}{\frac{s}{\sqrt{n}}}$ ~$t_{n-1}$?

Does $X_1,...,X_n$ being a random sample from $N(\mu,\sigma^2)$ $\implies \frac{\overline{X}-\mu}{\frac{s}{\sqrt{n}}}$ ~$t_{n-1}$? If so does the above imply that a standard normal divided by the ...
0
votes
1answer
18 views

Prove that for a fair coin, lengths of series of zeros or ones have geometric distribution

As in title: you're flipping a coin indefinitely (let's say heads gives 1, tails gives 0). E.g.: ${0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1\dots}$ How would you try proving that lengths of series of ...
5
votes
0answers
45 views

If $X\sim \mathrm{lognormal}$ then $Y:=(X-d|x\geq d)$ has approximately a Generalized Pareto distribution.

Let $X$ be a random variable with lognormal distribution. Show that when sufficiently large then $Y:=(X-d|x\geq d)$ is approximately a random variable with generalized Pareto distribution. Hint: Use ...
0
votes
2answers
26 views

Difficult demonstration - How to show that $H_n$ is normal distributed $N(\xi,\sigma^2)$ starting from its moments $ξ$ and $σ$?

I was thinking that if the function $H_n$ of cumulative distribution converges to a distribution $H$, then $\epsilon_n$ should converge to $\epsilon$ what could be expressed as follows: If $H_n$ is ...
0
votes
1answer
14 views

A example of a CDF tends in law to a variable Y but its variance does not tends to var(Y)

I was thinking in a example of Kn(Yn-c) tends in law to a ramdom variable Y with CDF(cumulative distribution function), but where Variance of Kn(Yn-c) does not tend to v^2=Var(Y). Do you think that ...
0
votes
2answers
16 views

How does the probability of events change if an event does not occur

Suppose that someone tells me I will collect $\$100$ dollars within some time interval. Those time intervals are 1 to 7 days, 8 to 30 days and eventually after 30 days. Let $A$ be the event I ...
2
votes
2answers
27 views

$U$~$N(3,16)$ $V$ ~$\chi_{9}^{2}$ U and V are independent random variables. Find $P(U-3<4.33\sqrt{V})$

$U$~$N(3,16)$ $V$ ~$\chi_{9}^{2}$ U and V are independent random variables. Find $P(U-3<4.33\sqrt{V})$ (The notes I'm working through don't seem to approach this rigorously...) The answer is $P(...
0
votes
1answer
23 views

Uniform distributed success probability for a coin

$n\in \Bbb N$. Let $X_1 \sim \text{Uni}_{(0,1)}$ and $X_2 \sim \text{Bin}_{n, X_1}$ conditional on $X_1$. I want to find the distribution function of the law of $X_1$ given $X_2 = k$, i.e. $\Bbb P (...
0
votes
0answers
12 views

Upper bound on number of cliques in a Vietoris-Rips complex

Does there exist an upper bound on the number of cliques of order $k$ in a Vietoris-Rips complex? I found this work --> https://arxiv.org/pdf/1104.0914.pdf I understand it makes the assumption of ...
2
votes
2answers
23 views

Convergence (to zero) for PDF of normal distribution.

I need to prove that the PDF converges to zero when $n\to\infty$; that is, $$\lim_{n\to \infty}f_n(x) =\lim_{n\to\infty} \frac{1}{\sqrt{2\pi n^{-3}}}\exp\left(-\frac{(x-\frac{1}{n})^2}{2n^{-3}}\right)...
0
votes
2answers
32 views

What will be the probablilty in these cases?

So we have a fair and unbiased dice, which is rolled thrice in a row. 1)What is the probability to get the sequence [1,2,3] in the three continuous trials? 2)What is the probability of getting the ...
0
votes
2answers
22 views

How is P(B) derived and why is $P(D_i)$ equal to 55/72 and not $(55/72)^i$

So this is a question with its solution below to which I don't understand 2 things. How is P(B) derived? And, why is $P(D_i)$=55/72 and not $(55/72)^i$. Since, for example, obtaining heads in the n ...
0
votes
1answer
29 views

How to setup the problem $P(Z^4 - 17 \ge 9)$ for Exponential(4) distribution

I'm working on part (d) of the problem that was discussed here on Math.StackExchange. The answer to that problem makes sense to me, and I've been able to use that answer to other, similar problems ...
1
vote
1answer
34 views

If $\|x_n\|_2 \to \infty$ in $L_2$-norm, does $\|x_n\|_{1+\varepsilon} \to \infty$ in $L_{1+\varepsilon}$-norm, for all $\varepsilon > 0$?

Question: If $\|x_n\|_2 \to \infty$ in $L_2$-norm, does $\|x_n\|_{1+\varepsilon} \to \infty$ in $L_{1+\varepsilon}$-norm, for all $\varepsilon > 0$? Details/Progress: This should follow trivially ...
0
votes
0answers
15 views

Variance of linear combination

This is a follow up question to this. Let $(X_1,\ldots, X_n)$ be non-independent random variables such that $$\sum_{i=1}^{n} X_i\sim\sum_{i=1}^{n} \alpha (\mathcal{N}(0,1))^2$$ where $\mathcal{N}(0,1)...
0
votes
1answer
21 views

Geometric distribution, can this approach be correct?

I am revising and came upon these questions, but I am not sure if the answers are right especially for part B Suppose the probability of defective is p(D) = 0.009 , p(d^c)=0.991 and mean 7 and std = ...
0
votes
0answers
3 views

How to transport a unique multimodal log-normal distribution by a set of moments?

I am using a CFD-QMOM approach to calculate the transport of nano particles through a fluid domain. The procedure that I am using is basically what is shown in this image: So I start with a certain ...
0
votes
1answer
51 views

Why is the standard deviation described as $\sqrt{pqn}$ sometimes and sometimes as $\sqrt{\frac{pq}{n}}$?

I assume it has something to do with whether we start with a distribution or with samples, but why is the standard deviation increasing with $n$ in one case and decreasing in the other?
1
vote
2answers
27 views

What is p(Evidence) exactly in a bayesian model?

I'm having a hard time intuitively understanding what this means in a machine learning context. When using the variables $A$ or $B$ or some trivial example, it all makes sense, but when looking at ...
0
votes
1answer
37 views

Compute the distribution function when $n=2$ and $n=3$

Two players $A$ and $B$ play a series of games that ends when one of them has won $n$ games. Suppose that each game played is, independently, won by player $A$ with probability $p$. Let $X$ be the ...
2
votes
1answer
23 views

How to model probability $x$ populations have recovered by day $k$?

Suppose the following scenario. $N$ patients are treated with an antibiotic, bringing the population of a certain bacteria in their guts down to negligible levels. I have the following 5 data points: ...
0
votes
1answer
36 views

Find a consistent estimator for $E[X^2]$ when $X \sim \text{Exp}(\beta)$

I am working on this problem. Find a consistent estimator for $E[X^2]$ when $X \sim \text{Exp}(\beta)$ . So far I am thinking of using the invariant property of MLEs, so I let $$\hat{\theta} = \...
1
vote
0answers
23 views

Visual Intuition: Gaussians closed under addition

I'm trying to develop some intuition for the fact that the family of Gaussian distributions is closed under addition. I.e. if $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$, then $Y = \sum_iX_i$ is also ...
0
votes
1answer
16 views

On average, 3 oil tankers arrive at a port each day, The time between arrivals follows an exponential distribution.

(a) Find the probability that the next tanker does not arrive until at least two days from now. (b) Find the probability that seven tankers arrive in one day. for first question we have the ...
1
vote
1answer
31 views

Does $|X_n|\le\Delta_n+\delta$, with $\Delta_n\overset{p}\to0$ imply $|X_n|\overset{p}\to0$?

Suppose $\Delta_n\overset{p}\to0$, and for any $\delta>0$, we have $$|X_n|\le\Delta_n+\delta.$$ Can we conclude that $$|X_n|\overset{p}\to0?$$ Here $\Delta_n\overset{p}\to0$ means that for any $\...
0
votes
0answers
21 views

If $\sigma$ is a scale parameter, i.e. $f_\sigma(x)=\frac{1}{\sigma}f_1(\frac{x}{\sigma})$ then $\bar X/\sigma$ is a pivotal quantity

This was taken from Casella-Berger. In page 427, it is asserted that, if $X_1\dots,X_n$ is a random sample of a population with density $f_\sigma(x)=\frac{1}{\sigma}f_1(\frac{x}{\sigma})$, then the ...
1
vote
2answers
41 views

Fitting experimental data to a theoretical probability distribution

This is based on an experiment I did with a standard pack of playing cards. From the pack of 52 cards, a set of six cards was selected at random (i.e. the first six cards at the top of the pack after ...
2
votes
1answer
49 views

Deriving distribution from conditional distribution

Hi guys I am having problems deriving $P(X = k)$ if $P(X = k|X+Y = n)$ = ${n}\choose{k}$ $\times$ $2^{-n} $ X and Y are i.i.d. random variables with values in $\mathbb{N_0}$. After playing a bit ...
0
votes
0answers
9 views

Mean and variance of conditional mean and variance

I was given two discrete r.v. $X$ and $Y$. I know how to compute $E(X|Y=y)$ and $V(X|Y=y)$ and i realize how you could treat $E(X|Y)$ and $V(X|Y)$ as random variables them self. Yet I'm ...
-2
votes
0answers
33 views

Finding the probability distribution using expected value and variance [on hold]

A random variable Y can only take values in {−3, 0, 3}. The expected value of Y is 0 and its variance is 4. How would you find the probability distribution of Y? Thanks for the help.
0
votes
0answers
14 views

how do i approach this question? do I calculate the overall probability? [duplicate]

Mysterioso the Magician is walking down the street with a box containing 25 identical looking coins: 24 are fair coins (which flip heads with probability 0.5 and tails with probability 0.5) and one is ...
2
votes
1answer
28 views

Given PDF find Mean of the distribution

pdf: $ f(x) = \frac{e^{-x}}{(1+e^{-x})^2} $ Find the mean of the distribution using: $ \int_{-\infty}^{\infty} xf(x) \ \mathrm{d}x $ Is it possible? I have run the integral in a calculator only ...
0
votes
1answer
21 views

conditonal distribution question

For conditional distribution $$f_{X|Y}(x|y) = \frac{f(x,y)}{f_Y(y)}$$ this is the basic definition I know about conditional distribution Consider n + m trials having a common probability of ...
0
votes
1answer
26 views

Does $\xi_{ni}\overset{p}\to0$ imply $\frac1n\sum_{i=1}^n\xi_{ni}\overset{p}\to0?$

Suppose for each $i=1,\cdots,n$, we have $$\xi_{ni}\overset{p}\to0.$$ Can we claim that $$\frac1n\sum_{i=1}^n\xi_{ni}\overset{p}\to0?$$ Here $\overset{p}\to$ means convergence in probability. That ...
0
votes
0answers
17 views

Given a probability density function $p$, find a factorized distribution $q$, such that $\frac{p(x)}{q(x)}$ is bounded above?

Given a bounded, continuous probability density function $p$ over $\mathbb{R}^n$, is it always possible to find a factorized distribution $q$, $q(x) = \prod_{i=1}^n q(x_i)$, so that $\frac{p(x)}{q(x)}$...
0
votes
1answer
18 views

Ways to form n using the elements in [k] without repetition

Is there a closed formula for $f(n,k)$ where $f(n,k)$ is equal to the number of different ways to sum the elements from $[k] = \{0,1,2,3,...k\}$ without using the same element more than once to form $...
0
votes
1answer
25 views

What is the probability of a tennis player winning both, or one of two sets?

Let's suppose we are calculating tennis, where two sets are played. The probability of player "A" winning the first set is 50% The probability of player "A" winning the second set is 50% (regardless ...