Questions tagged [geometric-distribution]

The geometric distribution is a discrete (count) distribution, where the probability of each count is a constant proportion of the next lower count. An example is 'the number of coin tosses until the first head'.

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Number of trials to observe an inconsistent outcome in Bernoulli distribution

Suppose I have a random variable $X$ following a Bernoulli distribution with some known probability $p$. How do I calculate the mean number of trials $N$ needed such that you'd expect to observe an ...
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How to find the average distance between randomly distributed points in a rectangle?

Assume there are n points randomly distributed in a rectangle (x being the height y being the width) shown below in the figure. I would like to calculate the average distance between 2 random red ...
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Probability exercise using geometric distribution---Roulette gambling

A gambler plays roulette at Monte Carlo and continues gambling, wagering the same amount each time on “Red”, until he wins for the first time. If the probability of “Red” is 18 and the gambler has ...
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Can you write a Geometric random variable as some combination of Bernoulli random variables?

Background Given $Y \sim \text{Binomial(n,p)}$, we can write $Y = \sum_{i=1}^{n} X_i$ where $X_1,X_2,...,X_n$ are iid $\text{Bernoulli}(p)$. This is useful in, for example, determining the mean of a ...
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Which variation estimate to prefer: SD, geomSD, CV?

I have a case where we see regional disparities in median received services. Services is a lognormally distributed variable. Plot is based on a lognormal model, examining y and sigma. You can see that ...
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Information about the parameter of a Geometric distribution

This may be pretty basic, but essentially I need to know how would the parameter, say p, of a geometric distribution be characterized with relation to the pmf? Is it a scale parameter? Is it a shape ...
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Test statistic for the geometric distribution

I start with a sampling distribution of an unknown discrete probability distribution with PMF $g(x)$. I want to test if this distribution is in fact a geometric distribution, this is my $H_0$. ...
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can you look this problem and solution?

A company produces IC (integrated-circuit) chips. (a) The produced chips are tested one at a time until a good chip is found. If the probability that at least three tests are needed equals 0.0225. ...
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can you guide me how can ı solve this?

Missiles are launched until one successfully reaches the target. If the expected number of launches is 2.5, find the probability that at most 3 attempts will be needed.
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Intuition behind memoryless process and geometric series

I was reading this problem (Page 6, THE BASKETBALL PROBLEM, MEMORYLESS PROCESSES AND THE GEOMETRIC SERIES) and stumbled upon the solution using the memoryless property. I cannot understand the ...
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Sample log geometric distribution from log probability

I want to sample from the geometric distribution for a very small success rate. The success rate is so small that I represent it by its log. I want the result to also be represented by its log. Is ...
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Beta-Geometric Posterior Predictive distribution via Empirical Bayes

My question is on deriving posterior predictive distribution for Beta-Geometric distribution in Empirical Bayes context. Let Geometric Survivor Function: $$S(t|\theta) = (1-\theta)^t$$ Let $\theta$ ...
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What does $E(X = k)$ mean if $X \sim Geom(p)$?

So, I have the following question for homework assignment: $X_{1}$, $X_{2}$ have geometric distributions with parameters $p$ and $1-p$ respectively, $X_{1} \sim Geom(p)$, $X_{2} \sim Geom(1-p)$. $X_{1}...
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Estimating the blockchain mining time for $N$ nodes

I am trying to simulate a set of times for the below problem. There are N nodes. Each node generates a random number($R$) in the range $[0,K]$ per second. Guess the time it takes by each node to ...
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Is it possible to view sequential independent trials as pre-determined with unknown outcome?

This question is best represented by the following short story: Alice and Bob have been captured and imprisoned on an island by an evil adversary. Each day they are captured there the jailer rolls a ...
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How many trials are needed to have 99% of all possible successes between n independent elements which can have at most 2 successes each?

I want to calculate the expected time of growth of a group of plants in a videogame. If I have $n$ independent plants and each plant has a probability $p$ of being picked each unit of time $t$ (or ...
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Based on the record X1 ,…, Xn what is the unbiased estimation of 1/p [duplicate]

If we investigate $n$ patients for SARS. The indicator of sequence of the trails is $X_i$ ($X_i=1$ is for success and $0$ is not success). And the sequence indicator is available for all n independent ...
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Flat “geometric distribution” by varying the probability of the Bernoulli trail

In a simulation I am working on, each day (time step) there is a chance that a condition changes (at which point it is stuck in the changed condition). Setting this probability to a fixed value (say 5%...
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Inferring an approximate distribution for noising of data given 300,000 samples of human noising [closed]

I'm trying to find a statistical way to get an approximate distribution of all human noising. I have a dataset of over 300,000 samples of people noising words. I took basic Statistics and I would know ...
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Let N be the number of times you roll a 6-sided die until you roll a 1. Let M be the sum of rolling N six-sided dice. What is the pdf of M?

I understand that a geometric distribution can be used to determine the pmf for N, but am lost on finding the distribution for M. Also, if done in reverse order: Let N be the result of rolling a 6-...
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Geometric distribution: finding canonical link and proving it is part of the natural exponential family?

Looking for some help on my statistics homework question! The background to the question is: suppose that you toss a biased coin repeatedly (and independently) until you get a head. Let Y denote the ...
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Proper length of random nonce for hash calculation (blockchain)?

I have a string s and I need to calculate a nonce such that when appending the nonce to the string, the generated hash starts with a given sequence. The hash has 256 bits, so the number of ...
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R - Geometric Distribition

Why pgeom(20,0.01,lower.tail = FALSE) 1-sum(dgeom(1:20,0.01)) produce different results in R? The result should be the same (...
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Expected number of tosses until winner in game with two players with two different coins

Two players each have a coin which gives heads with probability $0.7$ (player 1) and $0.3$ (player 2), respectively. Player 1 goes first, and the players alternate until someone gets heads. What ...
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Geometric standard deviation on a shifted mean

I want to compute upper and lower control limits for a benchmark value, based on a data series, using geometric mean and a multiple of geometric standard deviation, but am not confident about the ...
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ANOVA-like analysis for geometric distribution

I have a dataset (>10,000 samples total) that strongly appears to be geometrically distributed. For this dataset I have a way of partitioning that makes sense theoretically and I would like to know if ...
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Does the 10% condition apply to geometric distributions?

I know that the 10% condition is required to assume independence in binomial settings. However, in a geometric setting where there is no fixed n-value, how would one determine independence? I am a ...
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When (if ever) is the sum of two dependent geometric RVs negative binominal?

Imagine you have two random variables $X $ and $Y$, you know $$ X \sim \text{Geometric}(p) \\ X + Y \sim \text{Negative Binomial}(2, p) $$ I am interested in what if anything can be said about the ...
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If Distribution is Geometric does that mean underlying probability of success for each trial p is fixed?

we know that if p (probability of success at each trial is fixed then the probability of each trial, then probability of first success at kth trial is given by Geometric Distribution I need to ...
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Strange connection between Bernouilli, Uniform and Geometric distributions

Final update on 11/29/2019: I have worked on this a bit more, and wrote an article summarizing all the main findings. You can read it here. Let us consider $Z = X_1 + X_1 X_2 + X_1 X_2 X_3 +\cdots$ ...
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Given a geometric random variable, why is it that $P(X > x) = (1 - \theta)^x$ and not $\theta(1 - \theta)^x$?

Suppose $X$ ~ $Geometric(\theta)$, where the distribution is based on identically independent Bernoulli trials (each trial has $\theta$ = probability of a success). The distribution is for the number ...
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Gaps in histogram for geometric distribution

I'm attempting an assignment in which we're supposed to write a function to simulate a geometric distribution with $p=0.03$. While plotting a histograms for about $100000$ simulations of the function, ...
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How to work out the expected value of $X^3$ for a geometric distribution

I have scoured all my textbooks and all through the internet but haven't been able to find a suitable answer to this. I have struggled my way through finding the expected value of $X^2$, with the help ...
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Variance for events occurring with gamma and geometric distribution

I am presented with a problem as follows: A listener is receiving messages with a wait time in between two consecutive messages that is exponentially distributed with a mean of 1 time unit. After any ...
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Proving that $\lfloor X \rfloor /n$ is a random variable and identifying its distribution. From Probability B. Fristedt

Let $X$ have the exponential distribution. For $n=1,2,\ldots$ let $Y_{n}$ equal $\lfloor X\rfloor /n$. Prove that $Y_{n}$ is a random variable, and calculate and identify its distribution function. ...
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Geometric distribution with a capped number of trials - finding expectation and prior predictive distribution

So I am modeling a random variable which follows a geometric distribution with probability $\theta$ except that the total number of trials is capped at some value $n$. I.e., the probability mass ...
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Interpretation of cdf of geometric distribution

Given a geometric random variable $X$ with $p = 0.05$, I want to find (for example) $P(X \gt 10)$. Trivially, this is $1 - P(X\leq10)$, which can be evaluated with the cdf as $1-0.4013$ or $0.5987$. ...
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Geometric distribution with multiple trials

I was looking into geometric distributions to find the probability of the first success of some random variable X. So if p = 0.04, the geometric distribution looks something like this: I understand ...
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Sum of a random number of r.v.'s [closed]

A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, ...
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Geometric distribution with multiple success state and markovian succes probability

Let $X_t$ be a random process in the space $E:=\{F, S_1, S_2, S_3\}$ for each $t$. We can see it like it is a game where we can win in three different ways or we can fail. We play until we fail for ...
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Geometric distribution described with rate parameter

I don't understand this sentence from this paper (around equation $5$): The function $H(\tau)$ is the hazard function. $H(\tau) = \frac{P_{\text{gap}}(g = \tau)}{\sum_{t=\tau}^{\infty} P_{\...
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UMVUE- geometric distribution where $X$ is the number of failures preceding the first success

$X_1, \dots, X_n$ iis geometric: $P(X=x) = (1-p)^{x}p$, $x=0,1,2, \dots$ My Attempt: $T=\sum_{i=1}^n X_i$ is a sufficient statistic $W= \begin{cases}1 & X_1= 0,\\ 0 & X_1\neq 0\end{cases}$ ...
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Probability of $X_1 \geq X_2$

Suppose $X_1$ and $X_2$ are independent geometric random variables with parameter $p$. What is the probability that $X_1 \geq X_2$? I am confused about this question because we aren't told anything ...
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What can we say about $N_{i}$ where $N=N_{1}+\cdots+N_{m}$, $N\thicksim Geom(\frac{1-p}{p})$ and conditional distribution of $N_{j}$ is binomial

Suppose that the number of events $N$ is a Geometric random variable with mean $\frac{1-p}{p}$. Further suppose that each event can be classified into one of $m$ types with probabilities $p_{1},p_{2},\...
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Outliers on discrete data

Is there any robust methodology to identify outliers in the discrete data distribution. I am specifically concerned with discrete geometrical distribution? P.S. Data transformation does not seem to ...
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Difference between geometric distribution and negative binomial distribution

How do I differentiate between a problem of geometric distribution and that of Negative Binomial Distribution? Both include something around first success or failure. I'm confused.
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Probabilistic user behavior markov models on web

I am considering the following probabilistic Markov model of actions of a user on the results page of a search engine. The user examines the first result, with a probability $A$ he is satisfied with ...
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Bayesian posterior for Geometric Distribution

I have the following homework problem I am trying to solve for but am stuck with the posterior part. Note the the geometric distribution is a discrete distribution that has a probability mass function ...
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Unbiased estimator of p in geometric distribution

The answer to this question given by my professor was statistic T(x)= 1when X=0 and T(x) = 0 otherwise. Can I consider E(x) = (1-p)/p and then cross multiply and take 1/(1+x) as an unbiased estimator ...
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I think there is a little mistake in this exercise about the memoryless of Geometric Distribution

An exercise of Jacod and Protter: Let $X$ be Geometric. Show that for $i, j > 0$, $$P(X > i + j | X > i) = P(X > j)$$ I did it and I got a different asnwer: $$P(X > i + j | X > i) = ...