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Turning Nearest Neighbour Distribution of Poisson Scatter Theorem to Rayleigh Distribution By Multiplying Constant

Okay, I just realized I needed a different approach. This is a common problem shown for undergrads studying probability theory, so instead of deleting the question I will just answer it myself. The ...
BurgerMan's user avatar
0 votes

Cumulative Distribution function of a Poisson distribution in terms of its parameter.

Yes, $p_0(\lambda)$ is there, so the telescope sum is $p_0-p_n$. Only it drops out in the final formula, as the derivative/integral of $p_0(\lambda)$ is $-p_0(\lambda)$, so the final formula is OK.
Zalan Heszberger's user avatar
1 vote
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Determine the distribution of $X$

The law of total probability (LoTP) helps you compute the probability of an event by summing over simpler pieces, where the conditional probability of each piece is either given or is easy to compute. ...
Michael's user avatar
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6 votes

43 cookies are randomly given to 10 children. What's the probability each child receives at least 2 cookies?

Randomly distributing $m$ cookies to $n$ children, one cookie at a time, let $f(a,b,r)$ be the probability that each child ends up with at least two cookies, assuming There are $r$ cookies remaining ...
quasi's user avatar
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4 votes

43 cookies are randomly given to 10 children. What's the probability each child receives at least 2 cookies?

General case: Having $m$ cookies, $n$ children, and minimum number of $r$ cookies for each child, the probability $P_r(m,n)$ is given by $$\color{blue}{P_r(m,n)=\frac{n!S_r(m,n)}{n^m}} \tag{1}$$ where ...
Amir's user avatar
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6 votes
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43 cookies are randomly given to 10 children. What's the probability each child receives at least 2 cookies?

This idea is legitimate. We can summarize the series calculation as finding the coefficient of $\lambda^{43}/43!$ in $$F(\lambda) = e^{\lambda}\left(\sum_{k\ge 2} e^{-\lambda/10} \frac{(\lambda/10)^k}{...
Misha Lavrov's user avatar
2 votes
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Sum of $Y$ i.i.d Binomial Random variables $(n,p)$, where $Y$ follows poisson distribution

Assuming that $X$ is independent of $Y_{i}$'s and $Y$ is independent of $X_{i}$'s , you can apply the following two identities, For any two random variables $Z$ and $W$ with finite expectation and ...
Mr.Gandalf Sauron's user avatar
-1 votes

Sum of $Y$ i.i.d Binomial Random variables $(n,p)$, where $Y$ follows poisson distribution

Please see here. In particular, the expectation follows from Wald's equation.
van der Wolf's user avatar
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2 votes

Find the probability that strictly $2$ people arrived in the first hour in a Poisson Process

Here $\lambda$ is not known, which on the face of it may seem like you don't have enough information, but in fact you do. In a Poisson process of rate $\lambda$, the number of arrivals in an interval ...
Especially Lime's user avatar

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