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统计代写| Geometric and Negative Binomial stat代写

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:

统计代写I Linearity of expectation

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4.3 几何和负二项式
我们现在介绍两个比较著名的离散分布,几何分布和负二项分布,并计算它们的期望值。

故事 4.3.1(几何分布)。考虑一系列独立的伯努利试验,每个试验都具有相同的成功概率 $p \in(0,1)$,试验一直进行到成功为止。令 $X$ 为第一次成功试验之前的失败次数。然后$X$ 具有参数$p 的几何分布;$ 我们用$X \sim \operatorname{Geom}(p)$ 表示。

例如,如果我们抛一个公平的硬币直到它第一次出现正面,那么在第一次出现正面之前的尾部数量分布为 Geom(1/2)。

要从故事中获取几何 PMF,请将伯努利试验想象为以单个 1(成功)结尾的一串 0(失败)。每个 0 的概率为 $q=1-p$,而最后的 1 的概率为 $p$,因此一串 $k$ 的失败后跟一个成功的概率为 $q^{k} p$。
定理 4.3.2(几何 PMF)。如果 $X \sim \operatorname{Geom}(p)$,那么 $X$ 的 PMF 是
$$
P(X=k)=q^{k} p
$$
对于 $k=0,1,2,\ldots$,其中 $q=1-p$。
对于几何级数的回顾),我们有第一个 $n+1$ 试验是失败的:

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