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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Existence of moment generating function [closed]
Give a example of discrete random variable for which mgf $M(t)=E\left(e^{tx}
\right)$ does not exist . I have tried with geometric(p) distribution when $(1−p)e^t≥1$
, the mgf does not converge. Is it ...
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Likelihood estimation with 2 samples from a geometric distribution
Context
Given there are 2 groups that can be modelled as a geometric distribution as follows:
\begin{align*}
f(x_i;p_1) &= p_1(1-p_1)^{x_i - 1} \; x_i = 1,2,... \; 0<p_1 <1 \\
f(y_i;p_2) &...
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Probability mass function of time of first head
This is an exercise from the probability book by Ross. This is not homework.
Using conditional probability and the distribution of sum of two geometric random variables, the probability comes out to ...
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Limiting value of the moment generating function
Suppose that the discrete random variable $X_{n}$has a geometric distribution given by
$$f_{X_n}(x_n)=P_n{(1-P_n)}^{x_n}$$ where $$x_n={0,1,2,3,}$$ and $$P_n\ =\ \frac{\lambda}{n}$$ for $0<\lambda&...
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Find the likelihood ratio test for to test $H_0$:$\log(1-\theta)=\eta_0$ against $H_1$:$\log(1-\theta)\neq\eta_0$
My approach is given below.
Let $X_1, \ldots, X_n \stackrel{\text{i.i.d}}{\sim} \text{Geometric}(\theta)$, so that
$$f_{\theta}(x) = \theta(1 - \theta)^x, \hspace{10mm} x \in \mathbb{N}.$$
Find the ...
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Is it possible to derive pmf from a CDF with no support given?
Given a CDF of random variable, $$F_X(x)=\sum_{j=1}^x \left(\frac{1}{2}\right)^j = 1-\left(\frac{1}{2}\right)^x$$,
I want to derive pmf of this random variable $X$. But there is no information about ...
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First Success distribution with geometrically decreasing success probability?
If a treatment has constant success probability of $p$ per trial, the probability of success on trial $k$ is given by the First Success Distribution.
$$P(K\!=\!k\,|\,p) = p(1-p)^{k-1} \qquad k = 1, 2,...
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Goodness-of-fit Tests
Continuing from my previous question here.
Furthermore, I intend to perform the chi-squared test and plot QQ-plots to test the hypothesis $H_0:\lambda=1$. I do not get to see the actual data though; I ...
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Minimax estimator for geometric distribution
I'm trying to solve this problem:
Let $X$ be a single sample from Geo($p$) where $p ∈ (0, 1)$. Find a minimax estimator for $p$ under the loss $L(p, δ(x)) = (p−δ(x))^2/
p(1−p)$ .
I'm trying to put ...
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Find the Fisher Information for geometric distribution
Given $X1,\dotsc,Xn \sim \mathcal{Geo}(p)$ , and I need to find the MLE and the CI for the MLE.
I found the MLE for this distribution, using the maximum likelihood function: $L(p;X) = (1-p)^(Xi-1) * p$...
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Mean of geometric distribution is odds?
Context: I mean the $P(X=k)=(1-p)^k p$ not the $P(Y=k)=(1-p)^{k-1} p$. Apparently the mean of the 1st kind of geometric is $\frac{1-p}{p}$ instead of $\frac{1}{p}$ for the 2nd kind of geometric. I ...
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Where does the article "Bayes or not Bayes" get its P values?
In the article Bayes or not Bayes, is this the question?, in the paragraph before Figure 2, the author says:
Let us suppose that we want to investigate whether the sex ratio in hypothetical mice ...
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Is this a geometric distribution problem?
Suppose a student starts with test A, and will proceed to test B, then test C if he passes.
The probability for the student to pass test A is 30%.
The probability for the student to pass test B is 20%....
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Geometric Distribution problem where p accumulates at each round for Beerio Kart!
My friends and I are hosting a Beerio Kart tournament and we are having trouble calculating a probability pertaining to it.
So in Mario Kart on the switch there are 56 courses and we are going to play ...
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Difference between geometric distribution expectation and 1 - failure with Binomial
I'm trying to understand a simple problem:
How many times you'd need to roll two dice to get two ones in a single roll.
One way I see this is as a problem the geometric distribution describes. You ...
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Non-informative prior of a geometric distribution [duplicate]
If we are given a standard geometric distribution $(1-p)^{x-1} p$, with $0<p<1$ what would be a suitable non-informative prior for this?
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What statistical distribution would best capture a set of Wordle outcomes?
Wordle is a simple and popular word game. It is based on an older game, and has recently gone viral and attracted attention. It is available in several languages and the original site (now owned by ...
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Statistical test for geometric distribution
Question
I have a sample of data (~250 values) which I think is geometrically distributed. Is there any statistical test that I can use to check if it is indeed geometrically distributed? Ideally ...
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Sufficiency for Truncated Geometric
Here is a deviant of a question I feel like I have seen several times on truncated exponentials and similar distributions for finding sufficient statistics:
Let
$$\mathbb{P}(Y=y)=\theta^y(1-\theta)^{\...
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Finding the distributions of the max and min of random variables from the geometric distribution
Let $X$ and $Y$ be independent random variables following the same geometric distribution, that is $P(X=k)=P(Y=k) = (1-p)p^k, k=0,1,\ldots,$. Let $U=min\{X,Y\}$, $V=max\{X,Y\}$,and $W=V-U$. How do I ...
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Geometric Distribution in R
I'm trying to solve a problem involving a Geometric Distribution with $p = 0.20$ and $x = 5$. I use the formula and R, but I get two different answers:
\begin{eqnarray*}
P(X = x) & = & p(1 - p)...
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Confused about geometric distribution [duplicate]
So I have a game that follows a geometric distribution.
There is a probability, $p$, of winning a round. If the player wins, he earns \$1,000. As soon as there is a win, the game ends.
If the player ...
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Geometric Distribution: Playing a Game with Cost
Suppose I have \$3,000. I play a game, for which I have a probability $p$ of winning.
I have to pay \$300 each time I play the game. If I win, then I earn a payoff of \$500. I can play the game as ...
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Bias correction for MLE of mean of geometric random variable
Parameter estimation [ edit]
For both variants of the geometric distribution, the parameter $p$ can be estimated by equating the expected value with the sample mean. This is the method of moments, ...
user318514
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A description of the mean of the Geometric Distribution - is it unorthodox or just incorrect?
I have a homework assignment where I'm asked to propose an estimator for the mean of a geometric random variable. This seemed simple enough, given that I've always understood the mean of the geometric ...
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Number of tries until Failure with n different independent Bernoulli experiment
I have 3(n) coffee machines in an office. I have a historical log of these machines, and I know in the last ten days(t) how many times they failed to make a coffee.
For example:
machine 1: 10 ...
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KS goodness of fit test result for geometric distribution low p-value [duplicate]
I am trying to test if the sampled interval between random events fits a particular geometric distribution, and am pretty lost as to what I'm doing wrong.
Assuming there's nothing wrong with the ...
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What is the expectation of $e^X$, where $X$ is a random variable with a geometric distribution?
if $X$ is a random variable with a geometric distribution how can I calculate
$$
E(e^X)
$$
I have no idea on how to do that.
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Concentration of sum of geometric random variables taken to a power
I am interested in techniques for showing the concentration of sum of $n$ iid geometric random variables $X_1, X_2, \cdots, X_n$ (number of trials until success), say with success probability $p = 1/2$...
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Accurately estimating the parameters for mixture of geometric distributions
Say we have an i.i.d. sample from a mixture of Geometric distributions:
$$
\begin{cases}
Geo(p_1) &w.p. \pi_1\\
Geo(p_2) &w.p. 1- \pi_1
\end{cases}
$$
Call the parameter set ${\theta}=(\pi_1,...
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Intution about the memoryless property of geometric distribution [duplicate]
I was watching the video about the memoryless property of geometric distribution. Here is an excerpt from the video.
$$\begin{aligned} P(X \geq x+y \mid X \geq x) &=\frac{P(X \geq x+y, X \geq x)}...
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Given a success on the kth trial, the probability of success on the mth trial
I know that this form of the Geometric distribution gives the probability that for a success probability $p$, the kth trial out of k trials is the first success.
$Pr(X=k) = (1-p)^{k-1}p$
My question ...
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Which of two parameters was chosen for a sample?
Problem: $X$ is chosen to be $\text{Geom}(p_1)$ with probability $\frac{1}{2}$ and is otherwise chosen to be $\text{Geom}(p_2)$. Given a sample of size $n$ from $X$, guess whether $p_1$ or $p_2$ was ...
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Equivalence of two state Markov chain and sampling via geometric distribution
Let $\mathcal T = \{1,2,\ldots,T\}$ denote the set of points in time, $S = \{0,1\}$ the state space, $X = (X_t)_{t \in \mathcal T} \in S^\mathcal T$ a time series, $\alpha = \mathbb P(X_{t+1} = 0 \mid ...
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Does density belong to exponential family?
$$f(x;\theta) = 2x\theta\exp({-x^2})\left( \frac{\exp({-x^2})}{1-\exp({-x^2})}\right)^{\theta\ - 1}\mathbb I_{(\mathbb R_{++})}(x) $$
with $\theta \in \mathbb R_{++} $
does $f(x;\theta)$ belong to ...
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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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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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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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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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