# 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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. (...
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### 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) =...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $\phi_{X}(t)=E[e^{itx}]=\... • 109 0 votes 0 answers 165 views ### method of moments estimator and confidence interval - Geometric Distribution Question : i got to the point(its correct) that :$\hat{\theta}_{mm}$=$\frac{1}{\bar{X}}Var(\hat{\theta}_{mm})$=$\frac{\bar{X}-1}{\bar{X}^3}$now I need to construct the Confidence Interval ... 0 votes 2 answers 51 views ### Checking my logic for a discrete Random Variable problem from Hoel Port & Stone, Intro Prob. Ch 3 #16 From Hoel, Port and Stone, Intro to Probability Theory, Chapter 3, exercise #16: Let X and Y be independent random variables having geometric densities with parameters$p_1$and$p_2$respectively. ... 4 votes 2 answers 186 views ### Find all$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$X\sim G(\lambda)$,... • 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 ... • 375 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 ... • 51 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 ... • 712 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$N$... • 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? (... • 2,033 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$...
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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 ...
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### 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. ...
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### 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): ...
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### 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 ...
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### 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 ...
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### 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) ...
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### 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? ...
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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 ...