Questions tagged [geometric-distribution]

For all questions that involve the geometric distribution in the context of probability, that is, the law of a random variable whose outcome is the number of attempts we need before a first success in repeated Bernoulli experiments.

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Mass Function for a Geometric distribution with two non-fail outcomes.

Let's say we roll a fair $d$-sided die, and if the die rolls $x$ or higher, we add 1 to the success count (let's define this as $h$) and roll again, but if we roll $y$ or lower, we subtract one from ...
Lee Davis-Thalbourne's user avatar
1 vote
0 answers
24 views

Branching Process confusion

I'm having an hard time at understanding how to explicitly calculate the distribution law of a branching process. Let's call $(X_{i,j})_{i,j\,\geq\,1}$ a double set of indipendent random variables, ...
Don Abbondio's user avatar
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33 views

Marginal PDF from Joint PDF of two Geometric Distribution

$X$ and $Y$ are two independent Geometric distributions with the parameter '$p$'. $$Z = X+Y$$ $$W=X-Y$$ Now I have found the joint PDF as below: $P_{Z,W}(i,j)=p^2(1-p)^{i-2}$ for $i>0$, $-i<j&...
pixiethepixel's user avatar
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23 views

Geometric Distribution density function

$X_1, X_2, ..., X_n$ are random sample with pdf $f(x;\theta)=\theta(1-\theta)^x, x=0,1,2,...$ based on that I see that it is the same with $X_i$ ~ $GEO(\theta)$ because the pdf of geometric ...
Ocean's user avatar
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3 votes
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61 views

Geometric Distribution in Simple Random Walk

My question is related to this question here. This says that if $S_{n}$ is a simple random walk (with steps $+1$ or $-1$ with probability $p$ and $q$ respectively) started at $S_{0}=1$, and if $T=\min\...
Blitzkrieg's user avatar
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43 views

Why is the distribution of $\min(X,Y)$ unique here?

Say we have 2 geometric random variables $X$ and $Y$ with parameters $a$ and $b$ respectively. I am wondering 2 things: why is the distribution of $\min(X,Y)$ the same for all geometric random ...
Princess Mia's user avatar
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2 votes
1 answer
60 views

Why does $\mathbb{P}[X_i=j]= (\frac 1 n)^{j-1} \cdot (\frac {n-i +1} 1)$ here?

I am asking this to understand the coupon collector's problem better. Say that we are trying to collect $n$ coupons, and every time we obtain a box of cereal, we get a coupon, which has an equal ...
Princess Mia's user avatar
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1 vote
1 answer
69 views

Why does the failure rate of a geometric distribution equal to p?

I know that the formal defintion of the probability of failure for a discrete random variable, $T$, is $$h(t) = \frac{P(T=t)}{1-P(T \leq t-1)}$$ This is all good, but I was told that for geometric ...
MathMeToDeath's user avatar
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1 answer
63 views

nth toss being head: independent event or sequence of events?

Given: The bartender start tossing a coin when each customer orders, and if the coin lands on a head, the customer will get a free drink. Assuming the probability of the coin landing heads is p. The ...
Mzq's user avatar
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4 votes
2 answers
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Prove $\sum_{k=i}^{n} {k-1 \choose i-1} p^i (1-p)^{k-i} = \sum_{k=i}^{n} {n \choose k} p^k (1-p)^{n-k}$

Prove that the following two summations are equal for any positive integers $i\leq n$, and any real number $p$ between $0$ and $1$: $$ \sum_{k=i}^{n} {k-1 \choose i-1} p^i (1-p)^{k-i} = \sum_{k=i}^{n} ...
Yuzhen Feng's user avatar
0 votes
2 answers
100 views

Randomly select a box from $10$ boxes, where $5$ balls are randomly distributed. Is the geometric distribution a suitable approach? $P(n) =p^nq$

$5$ balls have been randomly distributed among $10$ boxes. You randomly select one box, then reach inside to draw out one ball. The probability of successfully extracting a ball is denoted as '$p$.' ...
bananenheld's user avatar
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1 answer
69 views

How to calculate Geometric distribution expected value by Law of total expectation?

Could anyone help me with this problem, please? I have to calculate the expected value of the geometric distribution using the law of total expectation: $$E(X) = E[E(X | Y )]$$ I think I have to ...
user38473's user avatar
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236 views

Expected value of maximum of two D20 rolls

I'm trying to work through a similar version of this problem. The idea is that you have a fair, 20-sided die, and $n$ turns. You are trying to decide how many times to roll it (each roll costs a turn)...
Chuck Rak's user avatar
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1 answer
118 views

Best estimator for geometric distribution

I need an estimator for geometric distribution $\text{Geom}(p)$ that best fits my data $X_1, X_2,\ldots$ Is $\widehat{p} = \dfrac{1}{\overline{X}}$ the answer? Both MLE and method of moments yield ...
skipi's user avatar
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2 votes
1 answer
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Calculating covariance for 10 dice rolls

I have two variables, X and Y. I also have a dice. I roll the dice 10 times. X is defined as the number of times I get a result greater than 3, meaning 4,5,6 and Y is defined as the number of times I ...
Operation Star Wars's user avatar
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0 answers
34 views

Combining binomial probabilities of many independent, non-summable variables

TL;DR: How can I calculate a single number to capture the overall "rarity" of falling at percentiles L_k across a large set of ...
Mr. P's user avatar
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0 votes
1 answer
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Expected value - you roll a d6 untill you get a first non-6, then you win $6^{n-1}$ dollars

Consider this probability excercise: You roll a six-sided die. If you roll a six, you keep rolling until you roll a first non-six. Then you win $6^n$ dollars, where n is the amount of times you rolled ...
Zyx's user avatar
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Determining whether Z = min{X,Y} is geometrically distributed and finding the corresponding parameter. [closed]

Let the numbers p and q in the interval (0, 1) be given. Assume that X ∼ Geometric(p) and Y ∼ Geometric(q) are independent random variables. Show that Z = min{X,Y} is geometrically distributed and ...
Mohammed's user avatar
-1 votes
2 answers
83 views

Poisson Process: distribution for time until next arrival

The arrival of a subway follows a Poisson Process with rate $\lambda$ and each subway that arrives is full independently with probability $1-p$ such that the arrival of the next (non-full) subway you ...
Robert Leopold's user avatar
1 vote
1 answer
34 views

Finding the Limit in Probability of $X_p \sim \text{Geo}(p)$ as $p \to 0$

In studying for an upcoming qualifying exam, I have encountered the following problem: Suppose $X_p \sim \text{Geo}(p)$, which is to say random viable $X_p$ has PMF $$ \mathbb{P}(X_p = n) = p(1-p)^{n-...
YessuhYessuhYessuh's user avatar
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Could someone explain the concept behind the expected length of a second run in a sequence of bernoulli trials?

I am trying to self-educate on probability theory. I am now studying out of a book and was trying to do some exercises, however, I am stuck on this exercise (by now I know the answer but I am stuck on ...
strateeg32's user avatar
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0 answers
28 views

On PMFs of random variables

Question Consider a sequence of independent trials with probability of success $p$ at each trial. Suppose that $Y$ and $X$ are the times at which the first and second successes happen, respectively. (...
Ethan Mark's user avatar
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163 views

Expected number of dice rolls to get 2 sixes: not geometric?

The geometric distribution tells us that when you have independent events and you want to obtain the expected value till you have a success, you know that E[x] = 1/p. P(2 sixes in a row) = (1/6)(1/6) =...
Mario Diw's user avatar
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0 answers
17 views

Geometric Distribution Expectation Question

Products on a production line are defective with probability $0.1$, stochastically independently of each other. Let $Y$ be the total number of products which an inspector checks who stops when the ...
user avatar
1 vote
1 answer
45 views

Time to get a specific value when rolling an M-sided dice N times per second

If I roll an M-sided dice N times per second, what is the expected time elapsed until I roll a specific fixed value V at least once? And how long do I need to wait to be certain to a certain degree, ...
Alexander Torstling's user avatar
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0 answers
22 views

Shortcut to random variable S that follows a geometric distribution

I'm looking at algorithm L of reservoir sampling and 1 of the steps are as follows: Let p be the probability of success and S be the number of failures before the first success. Then, S follows a ...
Joel Lim's user avatar
2 votes
1 answer
29 views

Ratios of boys and girls via reciprocal of geometric distribution

Given the setting: In a world where everyone wants a girl child, each family continues having babies till they have a girl. What do you think will the boy to girl ratio be eventually? (Assuming ...
Mattiatore's user avatar
0 votes
1 answer
22 views

Given two geometric distributions, what is the chance that one comes before the other?

Say you have two events which have a given chance to happen on each trial. I know that for any one of the events, the number of trials before a success is described by the geometric distribution. What ...
Maxwell Tabarrok's user avatar
0 votes
1 answer
93 views

Use the characteristic function of a geometric function to derive $E[X]=\frac{1}{p}$

Suppose $X$ has geometric distribution $Geo(p)$, I want to use the characteristic function $\phi_{X}(t)=\frac{pe^{it}}{1-(1-p)e^{it}}$ to derive $E[X]=\frac{1}{p}$. I tried $\phi_{X}(t)=E[e^{itx}]=\...
Eric L.'s user avatar
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0 answers
165 views

method of moments estimator and confidence interval - Geometric Distribution

Question : i got to the point(its correct) that : $\hat{\theta}_{mm}$ = $\frac{1}{\bar{X}}$ $Var(\hat{\theta}_{mm})$ = $\frac{\bar{X}-1}{\bar{X}^3}$ now I need to construct the Confidence Interval ...
WalaWizon's user avatar
0 votes
2 answers
51 views

Checking my logic for a discrete Random Variable problem from Hoel Port & Stone, Intro Prob. Ch 3 #16

From Hoel, Port and Stone, Intro to Probability Theory, Chapter 3, exercise #16: Let X and Y be independent random variables having geometric densities with parameters $p_1$ and $p_2$ respectively. ...
James Rowell's user avatar
4 votes
2 answers
186 views

Find all $f$ such that $X\sim\mathcal{G}(\lambda) \;\Rightarrow\; f(X)\sim \mathcal{G}(\mu)$

I found a nice problem recently, but could not come up with a solution: Find all functions $f:\mathbb{Z}_{\geq 0}\to\mathbb{Z}_{\geq 0}$ such that for all $0< \lambda < 1$, if $X\sim G(\lambda)$,...
Noomkwah's user avatar
  • 441
1 vote
1 answer
32 views

How to find out the conditional expectance

Let $X$ be a random variable with $X\sim\mathrm{Geom}(1/3)$. Let $Y$ be another random variable with $Y\sim\mathrm{Bin}(n, 1/4)$ where $n$ is the value taken by the random variable X. I'm trying to ...
Tingo Hugo's user avatar
3 votes
1 answer
91 views

Geometric Random Variable Expectation/Variance

Looking to find the expected value/variance of the following geometric variable defined as having success rate p, where trials are taken until we have a success (which happens with probability p) or ...
John Li's user avatar
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2 votes
3 answers
318 views

Expected value of highest die roll

Daniel will roll a fair, six-sided die until he gets a 4. What is the expected value of the highest number he rolls through this process? It seems the expected number of rolls is 6. Can we reword ...
shrizzy's user avatar
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1 vote
1 answer
77 views

Variance of flipping a coin until getting heads

Suppose we toss a fair coin. Let $N$ denote the number of tosses until we get heads. What is $Var(N)$? My approach to this question is to find compute $𝐸[𝑁^2]- E[N]^2$. I got $E[N] = 2$ since $N$ ...
shrizzy's user avatar
  • 712
5 votes
2 answers
750 views

If population keep trying till they have girl child, what will be the probability of population having more girls than boys and vice versa?

I was solving this problem: In a world where everyone wants a girl child, each family continues having babies till they have a girl. What do you think will the boy to girl ratio be eventually? (...
Mahesha999's user avatar
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1 vote
2 answers
72 views

PMF of sum of independent RVs

Let $X$ and $Y$ be independent random variables, having the same probability mass function $$p_{X}(k) = p_{Y}(k) = p(1-p)^{k-1}, k = 1,2,...$$ Find the probability mass function of $X + Y$. Compute $...
Alborz's user avatar
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1 vote
1 answer
35 views

need information about this type of distribution

I came across a question like this: "Suppose a large hospital is interested in recruiting patients with a specific medical condition for a experiment. overall, 4 in every 10 patients vesting this ...
techie11's user avatar
  • 391
0 votes
1 answer
160 views

Clarifying a question on the number of Bernoulli trials

The following question is from Blitztein-Hwang, Introduction to Probability: Independent Bernoulli trials are performed, with probability 1/2 of success, until there has been at least one success. ...
vxek's user avatar
  • 197
1 vote
1 answer
61 views

Bounding fractional moments of geometric random variable

The following two bounds for a fractional moment of a geometric random variable $X$ with $\mathbb{P}\left[X = k\right] = p \left(1 - p\right)^k$ where $k \geq 0$ are given in this paper (on page 12): ...
M_F's user avatar
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1 vote
1 answer
136 views

Computing the expectation and using the derivation trick

I am reading Casella & Berger's Statistical Inference and trying to solve some of the exercises. Looking at the correction of exercise 2.20, the author used a method I don't understand, even ...
Nicolas's user avatar
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1 vote
1 answer
66 views

Geometric Distribution problem where p accumulates at each round

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 ...
Jared's user avatar
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0 votes
1 answer
66 views

Find the probability a student left their phone in class [closed]

The question is broken down in three parts, I know the answers to all three questions, however can only work out (ii). It would be much appreciated if someone can break down the working out for (i) ...
rh1710's user avatar
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0 votes
0 answers
22 views

Cumulative probability of two events with a geometric distribution

Suppose you have a lightbulb which has an average time to fail of 5 days. The failure can be modelled as a geometric distribution. What is the probability of 2 bulbs failing within seven days? ...
Arvind Sami's user avatar
1 vote
1 answer
178 views

Geometric distribution over a lifetime resulting from a Poisson distribution with exponential rate

Let's consider that some particles get created at rate $\beta$ following a Poisson distribution over a period $\tau$ i.e. if X is the number of created particles, then $X \sim Poisson(\beta)$. A ...
rubyOnSails's user avatar
1 vote
0 answers
41 views

Moment Generating Function of Y=2^(-X), where X ~ Geom(1/2)

My End Goal To compute the following Expected value: $$ \DeclareMathOperator{\E}{\mathbb{E}} \E \;\left(\frac1{\sum\limits_{i=1}^{B} 2^{-X_i}}\right)$$ where: $B$ is a constant (e.g., $B=4096$) $X_i \...
user1780424's user avatar
2 votes
1 answer
3k views

Asymptotic distribution of MLE of geometric distribution

I need to find the asymptotic distribution of the MLE of a geometric distribution. I know $\overline X$ goes as $N(1/p, (1-p)/(n p^2))$. Using the delta method MLE=$1/\overline X$ goes as $N(p, (1-p)/(...
revathi ananthakrishnan's user avatar