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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Determining whether a successive instances are conditional or independent

I hope I can appropriately describe the problem. I have an example data set with 610 individuals with made up analogy (it's not really about cars). I want to determine the number of people have x ...
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UMP test for poisson vs geometric

I have two hypotheses: $H_0\!: X\sim\rm{Pos}(1)$ $H_1\!: \rm{Geometric}(0.5)$ for any $\alpha$ in $(0,1)$ How can I construct a uniformly most powerful (UMP) test? I used the ratio under the $H_0$ ...
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Conditional pmf and mean for geometric rv

I have a geometric r.v. p(1-p)^(n-1), n = 1, 2, 3, 4, ... I was able to figure out the conditional mean conditioned on X > a. My answer to this is E[X] + a. I am pretty sure that is correct but if ...
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unbiased estimate of a geometric model

I was reading a book where in one of the section it shows how to find the unbiased estimate of a geometric model. This is from the book: Let x denote $n =1$ realisation from a geometric $Geom(\pi)$ ...
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Expected number of tosses till first head comes up

Suppose that a fair coin is tossed repeatedly until a head is obtained for the first time. What is the expected number of tosses that will be required? What is the expected number of tails that will ...
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Convergence of distributions implies convolution (with itself) converge?

If 2 distributions converge in the limit, then do their convolutions also converge? e.g. I can show that for the Geometric distribution with random variable $T$, the scaled version with parameter ...
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How to bound a probability with Chernoff's inequality?

In my class, we were given Chernoff's inequality as $$P(X\le -t) \le e^{(-(\lambda t - \log( E(e^{-\lambda x}))))}$$ $$P(X\ge -t) \le e^{(-(\lambda t - \log( E(e^{\lambda x}))))}$$ It says that to ...
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logarithmic calibration and geometric mean - citation search

I have a logarithmic calibration line: MachineSignal = -3.21 log(concentration) + 21.9 It seems obvious to me, that if I have a triplicate measurement of concentrations (measured indirectly through ...
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correct use of Negative Binomial with a Geometric distribution in a mixed model (glmmPQL)

I am trying to fit a NB GLMM with a gemoetric distribution. I have come across very little information on this form of regression. And would like some pointers/reasurance. some literature is ...
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conditional mean of a geometric RV

Say, there are three nodes: $S$, $R$, $D$. $S$ transmits to $R$, $R$ stores the packets, and later transmits to $D$. At any time, either $S$ or $R$ is selected to transmit according to some random ...
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cumulative distribution function , cdf problem

I cannot understand how step 2 transformed to step 3, anybody help me please ???
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Show that for a Geometric distribution, the probability generating function is given by $\frac{ps}{1-qs}$, $q=1-p$

Suppose that $X$ has a geometric distribution with probability mass function $P(X=x) = q^{i-1}p$, $i=1,2,...$ and $q=1-p$ Show that its probability generating function is given by $ ...
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Maximum Likelihood for shifted Geometric Distribution

Really struggling with this please help. Find MLE for p and c \begin{equation} \ {f}(x,p,c) = (1-p)^{x-c}p \end{equation} x=c,c+1,c+2,..... p is between 0 and 1 c is element of the integers I am ...
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What is the probability of winning this Jackpot Lottery?

I am working on some more exam practice questions, and just want to see if I am approaching this question correctly? Here is the complete question: 2,250 numbers are drawn at random without ...
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Designing an experiment: Geometric or Bernoulli data

I have some process that succeeds or fails with probability $p$. I can do distinct simulations to estimate $p$: Run $N$ simulations of a single process, record $N$ samples of a ...
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Geometric v. arithmetic mean

According to this extract of a paper posted on the web, the average return of a fair coin flip that pays 100% for heads and loses 50 percent for tails over 3 periods is 25 percent per period, while ...
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Comparison of waiting times to geometric distribution

I am analysing data taken from observing about one million people over 24 months. For each person, each month is classified as a "success" or a "failure". I am specifically interested in the ...
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How to find the distribution of the result of a compound experiment

I'm trying to find the distribution of data collected from a die-roll and coin-toss experiment. The experiment is as follows: 1)Roll a fair die so that you get a number $D \in (1,...,6)$ 2)Flip a ...
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Distribution for first time when the value is less than the previous one

Let $X_i, i \geq 1,$ be independent uniform (0, 1) random variables, and define $N$ by $$N=\min\{n:X_n < X_{n-1}\}$$. I need to prove that $$P\{N \geq k | X_0=x\} = \frac{(1-x)^{k-1}}{(k-1)!}$$. I ...
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Expected value of modified geometric distribution

I am trying to find the expected value of $X$, where $X$ is the number of orders a customer will make in a lifetime. Assuming that there is a $p=.1$ chance of the customer placing an initial order, ...
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If $Y \sim geometric(P)$ and $P \sim \mathcal B(2, 1)$ how to compute $E(Y)$ and marginal pmf of $Y$?

$$Y \sim Geometric(P)\\ P \sim \mathcal B(2, 1)$$ I'm trying to compute $E[Y]$ without finding marginal distribution of $Y$. I need some hints here. I also need to find the pmf of $Y$. My approach ...
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Robust estimation of a geometric random variable

I have a bunch of data which is assumed to be instances of a geometric random variable with outliers. How can I do a robust estimation of the parameter $p$ so that the effect of outliers is minimized? ...
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Geometric distribution without replacement

On an attempt to solve this problem I've managed to reduce it to finding the expected number of white balls picked until one black ball is observed (let's call that value $v$). Except that, unlike the ...
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What is distribution of lengths of gaps between occurrences of ones in Bernoulli process?

Which distribution fits the following data? Data are generated by the process: $X_t, \, t=\{1,2,3,\ldots,n\}$ is equal 1 with probability $p$, and 0 with probability $(1-p)$ for each $t$. What is ...