### 统计代写|工程统计作业代写Engineering Statistics代考|Bayes’ Belief Calculations

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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|Bayes’ Belief Calculations

Belief is the confidence that you have in making a statement of fact about something, a supposition. Here are some examples of statements that we might make:

“These symptoms are just seasonal allergies.”
“The average benefit of Treatment $Y$ is larger than that of Treatment $X$.”
And for each we might claim to be very certain of the supposition (perhaps $99 \%$ sure), or somewhat certain (perhaps $80 \%$ sure), or even not sure whether it is or is not (perhaps $50 \%$ sure).

Bayes’ Belief, $B$, is scaled by $100 \%$, so the value is $0 \leq B \leq 1$. If you are not so certain about a statement, the belief, $B$, might be $0.25$. If you are very certain, $B$ might be $0.97$. If you are not so sure about something, and it could be a $50 / 50$ call, then $B=0.5$.

Because we act on suppositions, we want to be fairly certain that the statement about what we suppose represents the truth about the reality. When you are not certain, you perform tests, take samples, get other’s opinions, etc. to strengthen or to reject your belief in the supposition. But tests are not perfect. There is always some uncertainty about the results. For example, the manufacturer of a particular procedure for detecting the presence of colorectal cancer reports it detects the disease in $92 \%$ of the patients with cancer and gives a negative result in $87 \%$ of the patients without the disease. (Exact Sciences Laboratories, Cologuard Patient Guide, 2020). The $92 \%$ correct positives means $8 \%$ false negatives. (The test on $8 \%$ of patients with the disease will falsely indicate they do not have it.) Similarly, the $87 \%$ correct negatives means $13 \%$ false positives. (The test on $13 \%$ of patients without the disease will falsely indicate they have it.)
Table $2.1$ is a matrix of the probabilities of the medical test giving true and false indications.

Here is another example: A test for steady-state (SS) might look at the past several data points. At SS the time-rate of change, data slope, ideally is zero, $S=0$. But, because of noise on the data, the slope will not be exactly zero; so, you might accept SS if the test results are $-0.1 \leq S \leq+0.1$. So, if the test result indicates $S=-0.03$ you say that is just noise, and the test indicates SS. But, at SS, a particular confluence of data perturbations might indicate the local slope is $S=0.15$, and the test would reject the true condition of SS. Maybe, given a true SS, the test will indicate SS $85 \%$ of the time, and reject SS $15 \%$ of the time.

On the other hand, if the process is in a transient state (TS), the slope will be much greater than a SS value, the slope will be beyond the $-0.1 \leq S \leq+0.1$ limits, and the test result will claim TS. However, even in a TS when the process variable is rising, the noise pattern on the past few samples might have a decreasing pattern, and the rate of change might incorrectly indicate SS. Maybe, given a true TS, the test will indicate TS $95 \%$ of the time, and SS 5\%.

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

1. In flipping a fair coin twice, what is the probability of a) getting two Heads, b) getting two Tails, c) getting a Head on the first flip and a Tail on the second, d) not getting any Heads?
2. At a particular summer camp, the probability of getting a case of poison ivy is $0.15$ and the probability of getting sunburn is $0.45$. What is the probability of a) neither, b) both, c) only sunburn, d) only poison ivy?
3. After rolling three fair six-sided dice, what is the probability of a) getting three ones showing, b) having only one four showing, c) getting a one and a two and a three?
4. If the probability of rain tomorrow is $70 \%$ and rain the next day is $50 \%$, then $0 \%$ for the next five days, what is the probability of rain a) on both of the next two days, b) on all of the next seven days, c) at least once this week?
5. There are two safety systems on a process. If an over-pressure event happens in the process, the first safety override should quench the source, and if that is not adequate the back-up system should release excess gas to a vent system. Normal control of the process is generally adequate, only permitting an average of about ten over-pressure events per year. The quench system, we are told, has a $95 \%$ probability of working adequately when needed, and the back-up vent has a $98 \%$ probability of working as needed. What is the probability of an undesired event (the over-pressure happens, and it is not contained by either safety system) in a) the next one-year period, and b) the next ten-year period?
6. There is a belief that Treatment B is better than the current Treatment A in use. The belief is a modest $75 \%, B=0.75$. If $\mathrm{B}$ is equivalent to $\mathrm{A}$, not better, then there is a $50 / 50$ chance that the trial outcome will indicate either B is better or worse. However, if B is better, then the chance that it will appear better in the trial is $80 \%$. What is the new belief after the trial if a) the trial indicates B is better, b) if the trial indicates $B$ is not better, and c) how many trials of sequential successes are needed to make the belief that $B$ is the right choice raise to $99 \%$ ?
7. A restaurant buys thousands of jalapeno peppers per day, of which $5 \%$ are not spicy-hot. They use five peppers in each small batch of salsa. If two (or more) of the five are not hot, customers are likely to complain that the salsa is not adequate. What is the probability of making an inadequate batch of salsa? Quantify how larger batch sizes will change the probability.

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

Measurement: A numerical value indicating the extent, intensity, or measure of a characteristic of an object.

Data: Either singular as a single measurement (such as a $y$-value) or plural as a set of measurements (such as all the $y$-values). Data could refer to an input-output pair $(x, y)$ or the set $(x, y)$.
Observation: A recording of information on some characteristic of an object. Usually a paired set of measurements.

Sample: 1) A subset of possible results of a process that generates data. 2) A single observation.
Sample size: The number of observations, datasets, in the sample.
Population: All of the possible data from an event or process – usually $n=\infty$.
Random disturbance: Small influences on a process that are neither correlated to other variables nor correlated to their own prior values.

Random variable: A variable or function with values that are affected by many independent and random disturbances despite efforts to prevent such occurrences.

Discrete variable: A variable that can assume only isolated values, that is, values in a finite or countably infinite set. It may be the counting numbers, or it may be the digital display values of truncated data.

Continuum variable: A variable that can assume any value between two distinct numbers.
Frequency: The fraction of the number of observations within a specified range of numerical values relative to the total number of observations.

Cumulative frequency: The sum of the frequencies of all values less than or equal to a particular value.

Mean: A measure of location that provides information regarding the central value or point about which all members of the random variable $X$ are distributed. The mean of any distribution is a parameter denoted by the Greek letter $\mu$.

Variance: A parameter that measures the variability of individual population values $x_{i}$ about the population mean $\mu$. The population variance is indicated by $\sigma^{2}$.
Standard deviation: $\sigma$ is the positive square root of the variance.
Empirical Distributions: These are obtained from a sampling of the population data. As a result, the models or the parameter values that best fit a model to the data (such as $\mu$ and $\sigma$ ) may not exactly match those of the population.

Theoretical Distributions: These are obtained by derivation from concepts about the population. If the concepts are true, then the models and corresponding parameter values represent the population. But nature is not required to comply with human mental constructs.
Category (classification): The name of a grouping of like data, influences, events such as heads, defectives, zero-crossings, integers, negative numbers, green, etc.

## 统计代写|工程统计作业代写Engineering Statistics代考|Bayes’ Belief Calculations

“这些症状只是季节性过敏。”
“治疗的平均收益是大于治疗的X。”
“这个过程已经达到稳定状态。”

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

1. 抛一个公平的硬币两次，a) 得到两个正面，b) 得到两个反面，c) 第一次得到正面，第二次得到反面，d) 没有得到正面的概率是多少？
2. 在一个特定的夏令营中，得到一箱毒藤的概率是0.15晒伤的概率是0.45. a) 两者都没有，b) 两者都有，c) 只有晒伤，d) 只有毒藤的概率是多少？
3. 在掷出三个公平的六面骰子后，a) 得到三个 1，b) 只有一个 4，c) 得到 1 和 2 和 3 的概率是多少？
4. 如果明天下雨的概率是70%第二天下雨50%， 然后0%在接下来的 5 天里，a) 在接下来的两天中，b) 在接下来的所有 7 天中，c) 本周至少一次下雨的概率是多少？
5. 一个过程有两个安全系统。如果过程中发生过压事件，第一个安全超控应熄灭源，如果这还不够，备用系统应将多余的气体释放到排气系统。该过程的正常控制通常是足够的，每年仅允许平均约十次过压事件。我们被告知，淬火系统有一个95%在需要时充分工作的可能性，并且备用通风口具有98%根据需要工作的可能性。在 a) 下一个一年期和 b) 下一个十年期中发生意外事件（发生过压，并且两个安全系统均未包含）的概率是多少？
6. 有一种观点认为治疗 B 优于当前使用的治疗 A。信念是谦虚的75%,乙=0.75. 如果乙相当于一种，不是更好，那么有一个50/50试验结果表明 B 更好或更差的可能性。但是，如果 B 更好，那么它在试验中看起来更好的机会是80%. 如果 a) 试验表明 B 更好，b) 如果试验表明，试验后的新信念是什么乙不是更好，并且 c) 需要多少次连续成功的试验才能使人们相信乙是正确的选择99% ?
7. 一家餐馆每天购买数千个墨西哥胡椒，其中5%不辣。他们在每小批莎莎酱中使用五个辣椒。如果五个中的两个（或更多）不热，客户可能会抱怨莎莎酱不够。制作不足批次的莎莎酱的概率是多少？量化更大的批量将如何改变概率。

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