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# 统计代写| Discrete Uniform stat代写

## 统计代考

3.5 Discrete Uniform
A very simple story, closely connected to the naive definition of probability, describes picking a random number from some finite set of possibilities.

Story 3.5.1 (Discrete Uniform distribution). Let $C$ be a finite, nonempty set of numbers. Choose one of these numbers uniformly at random (i.e., all values in $C$ are equally likely). Call the chosen number $X$. Then $X$ is said to have the Discrete Uniform distribution with parameter $C$; we denote this by $X \sim \mathrm{DUnif}(C)$.
The PMF of $X \sim \operatorname{DUnif}(C)$ is
$$P(X=x)=\frac{1}{|C|}$$
for $x \in C$ (and 0 otherwise), since a PMF must sum to 1. As with questions based on the naive definition of probability, questions based on a Discrete Uniform distribution reduce to counting problems. Specifically, for $X \sim \mathrm{DUnif}(C)$ and any $A \subseteq C$, we have
$$P(X \in A)=\frac{|A|}{|C|} .$$
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Random variables and their distributions
Example 3.5.2 (Random slips of paper). There are 100 slips of paper in a hat, each of which has one of the numbers $1,2, \ldots, 100$ written on it, with no number appearing more than once. Five of the slips are drawn, one at a time.
First consider random sampling with replacement (with equal probabilities).
(a) What is the distribution of how many of the drawn slips have a value of at least 80 written on them?
(b) What is the distribution of the value of the $j$ th draw (for $1 \leq j \leq 5) ?$
(c) What is the probability that the number 100 is drawn at least once?
Now consider random sampling without replacement (with all sets of five slips equally likely to be chosen).
(d) What is the distribution of how many of the drawn slips have a value of at least 80 written on them?
(e) What is the distribution of the value of the $j$ th draw (for $1 \leq j \leq 5) ?$
(f) What is the probability that the number 100 is drawn in the sample?
Solution:
(a) By the story of the Binomial, the distribution is $\operatorname{Bin}(5,0.21)$.
(b) Let $X_{j}$ be the value of the $j$ th draw. By symmetry, $X_{j} \sim \operatorname{DUnif}(1,2, \ldots, 100) .$ There aren’t certain slips that love being chosen on the $j$ th draw and others that avoid being chosen then; all are equally likely.
(c) Taking complements,
$$P\left(X_{j}=100 \text { for at least one } j\right)=1-P\left(X_{1} \neq 100, \ldots, X_{5} \neq 100\right)$$
By the naive definition of probability, this is
$$1-(99 / 100)^{5} \approx 0.049$$
This solution just uses new notation for concepts from Chapter 1 . It is useful to have this new notation since it is compact and flexible. In the above calculation, it is important to see why
$$P\left(X_{1} \neq 100, \ldots, X_{5} \neq 100\right)=P\left(X_{1} \neq 100\right) \ldots P\left(X_{5} \neq 100\right)$$
This follows from the naive definition in this case, but a more general way to think about such statements is through independence of r.v.s, a concept discussed in detail in Section 3.8.
(d) By the story of the Hypergeometric, the distribution is HGeom(21, 79,5).
(e) Let $Y_{j}$ be the value of the $j$ th draw. By symmetry, $Y_{j} \sim \operatorname{DUnif}(1,2, \ldots, 100)$.

## 统计代考

3.5 离散均匀

$X \sim \operatorname{DUnif}(C)$ 的 PMF 是
$$P(X=x)=\frac{1}{|C|}$$

$$P(X \in A)=\frac{|A|}{|C|} 。$$
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(a) 有多少已抽出的单据上写有至少 80 的值，分布情况如何？
(b) 第 $j$ 次抽奖的价值分布是什么（对于 $1 \leq j \leq 5）？$
(c) 数字 100 至少被抽中一次的概率是多少？

(d) 有多少已抽出的单据上写的值至少为 80，分布情况如何？
(e) 第 $j$ 次抽奖的价值分布是什么（对于 $1 \leq j \leq 5）？$
(f) 在样本中抽取数字 100 的概率是多少？

(a) 根据二项式的故事，分布是 $\operatorname{Bin}(5,0.21)$。
(b) 令 $X_{j}$ 为第 $j$ 次抽奖的价值。通过对称性，$X_{j} \sim \operatorname{DUnif}(1,2, \ldots, 100) .$ 在第 $j$ 次抽签中，没有特定的纸条喜欢被选中，而其他纸条则避免被选中然后;所有的可能性都是一样的。
(c) 取补，
$$P\left(X_{j}=100 \text { 对于至少一个 } j\right)=1-P\left(X_{1} \neq 100, \ldots, X_{5} \neq 100\right)$$

$$1-(99 / 100)^{5} \约 0.049$$

$$P\left(X_{1} \neq 100, \ldots, X_{5} \neq 100\right)=P\left(X_{1} \neq 100\right) \ldots P\left(X_{5} \neq 100\右）$$

(d) 根据超几何的故事，分布是 HGeom(21, 79,5)。
(e) 设 $Y_{j}$ 为第 $j$ 次抽奖的价值。通过对称性，$Y_{j} \sim \operatorname{DUnif}(1,2, \ldots, 100)$。