### 统计代写|工程统计作业代写Engineering Statistics代考|Probability

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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|工程统计作业代写Engineering Statistics代考|Probability

An event is a particular outcome of a trial, test, experiment, or process. It is a particular category for the outcome. You define that category.

The outcome category could be dichotomous, meaning either one thing or another. In flipping a coin, the outcome is either a Head (H) or a Tail (T). In flipping an electric light switch to the “on” position, the result is either the light lights or it does not. In passing people on a walk, they either return the smile or do not. These events are mutually exclusive, meaning if one happens the other cannot. You could define the event as a $\mathrm{H}$, or as a T; as the light working, or the light not working.

Alternately, there could be any number of mutually exclusive events. If the outcome is one event, one possible outcome from all possible discrete outcomes, then it cannot be any other. The event of randomly sampling the alphabet could result in 26 possible outcomes. But if the event is defined as finding the letter ” $\mathrm{T}^{\prime \prime}$, this success excludes finding any of the other 25 letters.

By contrast, the outcome may be a continuum-valued variable, such as temperature, and the event might be defined as sampling a temperature with a value above $85^{\circ} \mathrm{F}$. A temperature of $84.9^{\circ} \mathrm{F}$ would not count as the event. A temperature of $85.1^{\circ} \mathrm{F}$ would count as the event. For continuum-valued variables, do not define an event as a particular value. If the event is defined as sampling a temperature value of $85^{\circ} \mathrm{F}$, then $84.9999999^{\circ} \mathrm{F}$ would not count as the event. Nor would $85.00000001^{\circ} \mathrm{F}$ count as the event. Mathematically, since a point has no width, the likelihood of getting an exact numerical value, is impossible. So, for continuum-valued outcomes, define an event as being greater than, or less than a particular value, or as being between two values.

One definition of probability is the ratio of the number of particular occurrences of event to the number of all possible occurrences of mutually exclusive events. This classical definition of probability requires that the total number of independent trials of the experiment be infinite. This definition is often not as useful as the relative-frequency definition. That interpretation of probability requires only that the experiment be repeated a finite number of times, $n$. Then, if an event $E$ occurs $n_{E}$ times out of $n$ trials and if the ratio $n_{E} / n$ tends to stabilize at some constant as $n$ becomes larger, the probability of $E$ is denoted as:
$$P(E)=\lim {n \rightarrow \infty} n{E} / n$$
The probability is a number between 0 and 1 and inclusive of the extremes 0 and 1 , $0 \leq P(E) \leq 1$.

## 统计代写|工程统计作业代写Engineering Statistics代考|A Priori Probability Calculations

Let us consider that $E_{1}$ and $E_{2}$ are two user-specified events (results) of outcomes of an experiment. Here are some definitions:

If $E_{1}$ and $E_{2}$ are the only possible outcomes of the experiment, then the collection of events $E_{1}$ and $E_{2}$ is said to be exhaustive. For instance, if $E_{1}$ is that the product meets specifications and $E_{2}$ is that the product does not meet specifications, then the collection $E_{1}$ and $E_{2}$ represents all possible outcomes and is exhaustive.

The events $E_{1}$ and $E_{2}$ are mutually exclusive if the occurrence of one event precludes the occurrence of the other event. For example, again, if $E_{1}$ is that the product meets specifications and $E_{2}$ is that the product does not meet specifications, then $E_{1}$ precludes $E_{2}$, they are mutually exclusive, if the outcome is one, then it cannot be the other.

Event $E_{1}$ is independent of event $E_{2}$ if the probability of occurrence of $E_{1}$ is not affected by $E_{2}$ and vice versa. For example, flip a coin and roll a die. The coin flip event of being a Head is independent of the number that the die roll reveals. As another example, $E_{1}$ might be that the product meets specifications, and $E_{2}$ might be that fewer than two employees called in sick. These are independent.

The composite event ” $E_{1}$ and $E_{2} “$ means that both events occur. For example, you flipped a $\mathrm{H}$ and rolled a 3. If the events are mutually exclusive, then the probability that both can occur is zero.

The composite event ” $E_{1}$ or $E_{2}^{\prime \prime}$ means that at least one of events $E_{1}$ and $E_{2}$ occurs. When you flipped and rolled, a H and/or a 3 were the outcomes. This situation allows both $E_{1}$ and $E_{2}$ to occur but does not require that result, as does the ” $E_{1}$ and $E_{2}{ }^{\prime \prime}$ case.
There could be any number of user-specified events, $E_{1}, E_{2}, E_{3}, \ldots, E_{n}{ }^{\circ}$ Two rules govern the calculation of a priori probabilities.

## 统计代写|工程统计作业代写Engineering Statistics代考|Conditional Probability Calculations

In some cases, an event has happened and we wish to determine the probability a posteriori (after the fact) that a particular set of circumstances existed based on the results already obtained. Suppose that several factors $B_{i}, i=1, n$ can affect the outcome of a specific situation or event, $E$. The probability that any of the $B_{i}$ did occur, given that the event or outcome $E$ has already occurred, is a conditional probability. Let’s begin with the premise that $B_{1}, B_{2}, B_{3}$, and $B_{4}$ can influence $E$, an event that has happened. The final event $E$ can take place only if at least one of the preliminary events (the $B_{i}$ ) has already happened. The probability that a particular one of them, e.g., $B_{3}$ occurred is $P\left(B_{3} \mid E\right)$. If one of the $B_{i y}$ say $B_{3}$, had to happen for $E$ to transpire, then $B_{3}$ is conditional on $E$.
These are end-of-process events and beginning-of-process conditions.
Conditional probabilities can be determined by the use of Bayes’ theorem. Bayes’ theorem is stated in Equation (2.10), where $P\left(B_{i}\right)$ and $P\left(B_{k}\right)$ are the a priori probabilities of the occurrences of events $B_{i}$ and $B_{k}$ and $P\left(B_{i} \mid E\right)$ and $P\left(B_{k} \mid E\right)$ are the conditional probabilities that $B_{i}$ or $B_{k}$ would occur if event $E$ has already occurred.
$$P\left(B_{k} \mid E\right)=\frac{P\left(B_{k}\right) P\left(E \mid B_{k}\right)}{n}$$
Here:

1. $E$ is an event, an outcome.
2. $B$ is a condition (a situation, or an influence).
3. $P(B)$ is the probability of a condition happening.
4. $P(E \mid B)$ is the probability $E$ occurring given that $B$ did.
5. $P(E) \cdot P(E \mid B)$ is the probability $B$ and it caused $E$.
6. $\Sigma$ is the sum of all probabilities of all ways that $E$ could happen.
7. $P(B \mid E)$ is the probability $B$ happening given that $E$ did.

## 统计代写|工程统计作业代写Engineering Statistics代考|Conditional Probability Calculations

1. 和是一个事件，一个结果。
2. 乙是一种条件（情况或影响）。
3. 磷(乙)是某种情况发生的概率。
4. 磷(和∣乙)是概率和鉴于发生乙做过。
5. 磷(和)⋅磷(和∣乙)是概率乙它导致和.
6. Σ是所有方式的所有概率的总和和可能发生。
7. 磷(乙∣和)是概率乙鉴于发生和做过。

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## MATLAB代写

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