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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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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

Not sure how to word this question, sorry if it's dumb. But 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 ...
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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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How much data is considered “sparse” for fitting a mixed (Beta Geometric) distribution with 4 shape parameters?

I'm using CamDavidsonPhillips Customer Lifetime Value library to calculate CLV, and it uses a distribution based on Peter Fader's work on the subject that fits a Gamma distribution to model customer ...
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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) = ...
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Distribution of number of heads by coin flipped a geometric number of times

I have random variables $N \sim \text{geo}(p)$, and $B | N \sim \text{bin}(N, q)$. I'm looking for the distribution of $B$. To be clear, I have $$ \mathbb{P}(N=n) = p(1-p)^n \quad \text{for } n = 0, ...
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Joint probability distribution of geometric distribution

Let $X$ and $Y$ be independent and identically distributed $(i.i.d.)$ r.v.’s, each having the probability distribution, $p(k) = (1 − λ)λ^k$; $k = 0,1,...$ where $λ :(0; 1)$ is a constant. Define $U = ...
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Evaluating usefulness of estimations of a parameter for different distributions

If I had a sample of size n and wished to estimate some parameter, say p for two different distributions from the produced sample what would be required to determine which was more useful? Assume the ...
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average number of tossing coins till 'Head-Tail-Head' series of observation comes up [duplicate]

I know that we use 'geometric distribution' to solve problems such as 'Expected number of tosses till first head comes up' or 'expected number of rolling dice till first 'number 6' comes up'. However,...
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Coefficient of variation (CV) of log-transformed data

I understand that with log-transformed data, the coefficient of variation (CV) on the original scale is equal to sqrt(exp(sigma^2)-1), where sigma is the standard ...
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Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers

Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers $Y_1, Y_2, ...$ from the uniform distribution on $[0, 1]$, until ...
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Sufficient and complete statistic function for $\theta$ of geometric distribution [duplicate]

I am trying to find a sufficient and complete statistics function for $0<\theta<1$ of a sample $X = X_1, \dots, X_n$ from the Geometric Distribution. We have $f(x;\theta) = (1-\theta)^{x-1}\...
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192 views

How would I calculate a combination of the Binomial and Geometric Distributions?

To be specific with my problem, I'm calculating a formula for a game. There's 6 independent trials, each with an independent probability of success = 0.34. I know that from the Geometric distribution, ...
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Distances between random points in a hypercube and statistics of exponents

TL;DR: Why is $\text{avg}\left(|a-b|^k\right)=\frac{2}{(k+1)(k+2)}$? I.e. for $k=2$, as for finding average Euclidian distances, the result is $\frac{1}{6}$? I've been reading a book about "Corobs," ...
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Geometric distribution with random, varying success probability

I'm really sorry if this question is too basic, but I've been looking for a while and haven't been able to find a convincing response. My statistics background is rather poor. Geometric distribution ...
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Geometric Distribution - Biased Coin Flip

Whilst studying I stumbled on this problem, which I wish to check if my understanding is correct. Imagine we have a biased coin with probability 'k' of getting a head when flipped. Now A defines the ...
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Maximum likelihood estimate for geometric distribution from table

I am asked to compute the MLE of the parameter p of the geometric distribution and then apply it to some given data. It is easy to find the MLE: But how do I apply it to these data: ...
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Geometric Distribution

I'm trying to solve the following problem: The number of bombs required to achieve the disintegration is assumed to be geometrically distributed. In one series in which y bombs are available, x of ...
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How random variable $Y$ following geometric distribution in the following situation

I have a random variable, $X$ which follows exponential distribution with parameter $\lambda$. Then, define $Y=k$ for some $a$ greater than zero such that: $ka \leq X \leq (k+1)a$. Now, my ...
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Negative Multinomial Distribution as a sum of random variables

Given a random variable $x$ distributed according to a Negative Binomial distribution with dispersion parameter $r$, it can be seen as the sum of $r$ Geometric distributions. But given the Negative ...
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What is the probability that a best of seven series goes to the seventh game with negative binomial

Why cant we use negative binomial to calculate the probability of getting a 7th game in best of 7 game series? P(X=7, r=4) = {7-1 C 4 -1} * (.5)^(7-4) * (.5)^4 = .15 , so there is 15 % chance of 7 ...
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Confidence interval using Central Limit Theorem

I've been trying to find this information online, but have not had much success so far. I want to approximate the 95% confidence interval for the geometric distribution with the following parameters: ...
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Constant Curve for Cumulative Binomial

I am simulating traffic flow and have come across a rather interesting statistics problem that I'm having trouble wrapping my head around. I am modeling a stretch of road comprised of 2 lanes. Lets ...
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The special case of the negative binomial, the geometric and calculation with scipy

A particular case? One might consider the geometric distribution is simply a special case of the negative binomial with $N=1$. We know the geometric provides: "The probability the first occurrence ...
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Proof that the floor of an exponential random variable is a geometric variable

For any real number $x$,$[x]$ represents the smallest integer greater than or equal to $x$. If $X$ is an exponential random variable with mean $1/K$,show that $[X]$ is a geometric random variable with ...
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CRLB for estimating $\theta$ of $\sim\text{Geo}(\theta)$

I'd like to ask if the below computation of the information number for the CRLB is correct: Consider $x_1, x_2$ as iid $\sim Geo(\theta)$ Since $x_1, x_2$ ae iid and the geometric distribution is ...
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419 views

Expansion of Cumulant Generating Function of Negbin

Let $X \thicksim Negbin(r,p)$ where $(0\lt p \lt 1) $ I want to derive skewness and kurtosis of $X$ by getting the Cgf of X. First, since Followance of Negative Binomial equals to the distribution ...
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Expected duration of a conflict

I'm currently struggling with some statistical work, and thus I'm seeking your advice. I'm trying to investigate the expected duration of a conflict in a) a democracy and b) an autocracy. What I do ...
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If $Y$ ~ $geo(1-\theta)$, what's a ML estimator for $\theta$?

Suppose we have $Y$ ~ $geo(1-\theta)$ for $\theta \in (0,1)$. That is the pdf of $Y$ is given by $\theta^{k-1}(1-\theta).$ Formally, from what I understand the ML estimator for $1-\theta$ comes out ...
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Convolution of random variables: unimodality of the likelihood function

Let $X_1, X_2,...X_k$ be random independent variables, each $X_i$ drawn from a Geometric distribution $\mathcal{G}(p_i)$, and let its convolution, or sum, be $Y = \sum_{i=1}^k X_i$. The likelihood ...
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Total distribution of m geometric distributions

Suppose that you made a webpage and you are collecting the statistics from the visitors. There are m types of visitors (students, company recruiters, etc.). Each visit is equally likely to be any of ...
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Convergence in probability and distribution

Let $P(K=k)=(1-\beta)^k\beta ; k=1,2,3,...$ Then it is required to show $\beta K$ converges in distribution to an exp (1) random variable as $\beta$ tends to zero. For this they have started with ...
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Relationship between the binomial and the geometric distribution

I want to know the relationship between binomial and geometic distribution. I know the distribution both have two outcome and probability of success is the same for both distribution.
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How to estimate retention rate with limited data?

Let's say I'm a gym owner and I have only 2 years of monthly data of my gym memberships. Some members have quit after a few months and some members have been a member for all 2 years (24 months). I ...
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Checking if two geometric distributions have the same mean [duplicate]

I have 2 geometric distributions. And I have to compare their means and get the p-value. Is there a statistical test specifically designed for geometric distributions? I already used Mann-Whitney for ...