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统计代写| Geometric and Negative Binomial stat代写

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4.3 Geometric and Negative Binomial
We now introduce two more famous discrete distributions, the Geometric and Negative Binomial, and calculate their expected values.

Story 4.3.1 (Geometric distribution). Consider a sequence of independent Bernoulli trials, each with the same success probability $p \in(0,1)$, with trials performed until a success occurs. Let $X$ be the number of failures before the first successful trial. Then $X$ has the Geometric distribution with parameter $p ;$ we denote this by $X \sim \operatorname{Geom}(p)$.

For example, if we flip a fair coin until it lands Heads for the first time, then the number of Tails before the first occurrence of Heads is distributed as Geom(1/2).

To get the Geometric PMF from the story, imagine the Bernoulli trials as a string of 0 ‘s (failures) ending in a single 1 (success). Each 0 has probability $q=1-p$ and the final 1 has probability $p$, so a string of $k$ failures followed by one success has probability $q^{k} p$.
Theorem 4.3.2 (Geometric PMF). If $X \sim \operatorname{Geom}(p)$, then the PMF of $X$ is
$$P(X=k)=q^{k} p$$
for $k=0,1,2, \ldots$, where $q=1-p$.
for a review of geometric series), we have the first $n+1$ trials were failures:

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4.3 几何和负二项式

$$P(X=k)=q^{k} p$$