### 统计代写|统计推断作业代写statistical inference代考|The Bernoulli distribution

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|统计推断作业代写statistical inference代考|The Bernoulli distribution

The Bernoulli distribution arises as the result of a binary outcome, such as a coin flip. Thus, Bernoulli random variables take (only) the values 1 and 0 with probabilities of (say) $p$ and $1-p$, respectively. Recall that the PMF for a Bernoulli random variable $X$ is $P(X=x)=p^{x}(1-p)^{1-x}$.

The mean of a Bernoulli random variable is $p$ and the variance is $p(1-p)$. If we let $X$ be a Bernoulli random variable, it is typical to call $X=1$ as a “success” and $X=0$ as a “failure”.

If a random variable follows a Bernoulli distribution with success probability $p$ we write that $X \sim \operatorname{Bernoulli}(p)$.

Bernoulli random variables are commonly used for modeling any binary trait for a random sample. So, for example, in a random sample whether or not a participant has high blood pressure would be reasonably modeled as Bernoulli.

## 统计代写|统计推断作业代写statistical inference代考|Binomial trials

The binomial random variables are obtained as the sum of iid Bernoulli trials. So if a Bernoulli trial is the result of a coin flip, a binomial random variable is the total number of heads.
To write it out as mathematics, let $X_{1}, \ldots, X_{n}$ be iid Bernoulli $(p)$, then $X=\sum_{i=1}^{n} X_{i}$ is a binomial random variable. We write out that $X \sim$ Binomial $(n, p)$. The binomial mass function is
$$P(X=x)=\left(\begin{array}{l} n \ x \end{array}\right) p^{x}(1-p)^{n-x}$$
where $x=0, \ldots, n$. Recall that the notation
$$\left(\begin{array}{c} n \ x \end{array}\right)=\frac{n !}{x !(n-x) !}$$
(read ” $n$ choose $x$ “) counts the number of ways of selecting $x$ items out of $n$ without replacement disregarding the order of the items. It turns out that $n$ choose $0, n$ choose 1 and $n$ choose $n-1$ are all 1 .

## 统计代写|统计推断作业代写statistical inference代考|The normal distribution

The normal distribution is easily the handiest distribution in all of statistics. It can be used in an endless variety of settings. Moreover, as we’ll see later on in the course, sample means follow normal distributions for large sample sizes.
Remember the goal of probability modeling. We are assuming a probability distribution for our population as a way of parsimoniously characterizing it. In fact, the normal distribution only requires two numbers to characterize it. Specifically, a random variable is said to follow a normal or Gaussian distribution with mean $\mu$ and variance $\sigma^{2}$ if the associated density is:
$$\left(2 \pi \sigma^{2}\right)^{-1 / 2} e^{-(x-\mu)^{2} / 2 \sigma^{2}} .$$
If $X$ is a RV with this density then $E[X]=\mu$ and $\operatorname{Var}(X)=\sigma^{2}$. That is, the normal distribution is characterized by the mean and variance. We write $X \sim N\left(\mu, \sigma^{2}\right)$ to denote a normal random variable. When $\mu=0$ and $\sigma=1$ the resulting distribution is called the standard normal distribution. Standard normal RVs are often labeled $Z$
Consider an example, if we say that intelligence quotients are normally distributed with a mean of 100 and a standard deviation of 15 . Then, we are saying that if we randomly sample a person from this population, the probability that they have an IQ of say 120 or larger, is governed by a normal distribution with a mean of 100 and a variance of $15^{2}$.

Taken another way, if we know that the population is normally distributed then to estimate everything about the population, we need only estimate the population mean and variance. (Estimated by the sample mean and the sample variance.)

## 统计代写|统计推断作业代写statistical inference代考|Binomial trials

(n X)=n!X!(n−X)!
（读 ”n选择X”) 统计选择方式的数量X出的物品n不考虑物品的顺序而无需更换。事实证明n选择0,n选择 1 和n选择n−1都是 1 。

## 统计代写|统计推断作业代写statistical inference代考|The normal distribution

(2圆周率σ2)−1/2和−(X−μ)2/2σ2.

## 广义线性模型代考

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## MATLAB代写

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