### 统计代写|工程统计作业代写Engineering Statistics代考|Values of Distributions and Inverses

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• 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|For Continuum-Valued Variables

For continuum-valued variables, $x$, the cumulative distribution function is the probability of getting a particular value or a lower value of a variable. It is the left-sided area on the probability density curve, often expressed as alpha. It is variously represented as $C D F(x)=F(x)=\alpha=p$. Here we ll use the $C D F(x)$ notation.

For continuous-valued variables, $x$, the probability distribution function, $p d f(x)$, represents the rate of increase of probability of occurrence of the value $x$. An alternate notation is $p d f(x)=f(x)$.

The relation between CDF(x) and $p d f(x)$ is
$$\operatorname{CDF}(x)=\int_{x_{\text {mimeme }}}^{x} p d f(x) d x$$
Where $x_{\text {minimun }}$ represents the lowest possible value for $x$. In a normal distribution $x_{\operatorname{minimum}}=$ $-\infty$. For a chi-squared distribution $x_{\text {minimun }}=0$.

The left-hand sketch in Figure $3.16$ illustrates the $C D F$ and the right-hand sketch the $p d f$ of $z$ for a standard normal distribution (the mean is zero and the standard deviation is unity). At a value of $z=-1$, the $C D F$ is about $0.158$, and the rate of increase of the $C D F$, the $p d f$ is about $0.242$. The notations are $0.158=C D F(-1)$ and $0.242=p d f(-1)$. In both you enter the graph on the horizontal axis, the $z$-value, and read the value on the vertical axis.

For continuous-valued variables the inverse of the CDF is the value of $x$ for which the probability of getting the value of $x$ or a lower value is equal to the CDF $(x)$.

The inverse would enter on the vertical axis to read the value on the horizontal axis. If the inverse question is, “What $z$-value marks the point for which equal or lower $z$-values have a probability of $0.158$ of occurring?” then we represent this inverse question as $z=\operatorname{CDF}^{-1}(\alpha)$. In this illustration, $-1=\operatorname{CDF}^{-1}(0.158)$. The inverse of the right-hand $p d f$ graph is not unique. If the question is to determine the $z$-value for which the $p d f=0.242$, there are two values, $z=-1$, and $z=+1$.

## 统计代写|工程统计作业代写Engineering Statistics代考|For Discrete-Valued Variables

For discrete-valued variables, $x$, likely a count of the number of events, the cumulative distribution function is the probability of getting a particular value or a lower value of a variable. It is the left-sided area on the probability density curve, often expressed as alpha. It is variously represented as $\operatorname{CDF}(x)=F(x)=\alpha=p$. Again, we will use the $C D F(x)$ notation.
For discrete-valued variables, $x$, the point distribution function, $p d f(x)$, represents the probability of an occurrence of the value $x$. An alternate notation is $p d f(x)=f(x)$. Here, $p d f(x)$ is a probability of a particular value of $x$, not the rate that the CDF is increasing. Unfortunately, the same symbol is used in continuum-valued distribution.
The relation between $\operatorname{CDF}(x)$ and $p d f(x)$ is
$$\operatorname{CDF}(x)=\sum_{x_{\text {meimum }}}^{x} p d f(x)$$

where $x_{\text {minimum }}$ represents the lowest possible value for $x$. Normally $x_{\text {minimum }}=0$, the least number of events that could occur.

The left-hand sketch of Figure $3.17$ illustrates the CDF and the right-hand sketch the $p d f$ of $s$, the count of the number of successes, for a binomial distribution (the number of trials is 40 , and the probability of success on any particular trial is $0.3$ ). Note that the markers on the graphs represent feasible values. The light line connecting the dots is a visual convenience. It is not possible to have $10.3$ successes. At a value of $s=10$, the CDF is about $0.309$, meaning that there is about a $31 \%$ chance of getting 10 or fewer successes. The $p d f$ is about $0.113$, meaning that the probability of getting exactly 10 successes is about $11 \%$. The notations are $0.309=\operatorname{CDF}(10)$ and $0.113=p d f(10)$. In both you enter the graph on the horizontal axis, the $s$-value, and read the value on the vertical axis.

For discrete-valued variables the inverse of the $C D F$ is the value of $s$ for which the probability of getting the value of $s$ or a lower value is equal to the CDF(s).
The inverse would enter on the vertical axis to read the value on the horizontal axis. If the inverse question is, “What $s$-value marks the point for which equal or lower counts have a probability of $0.309$ of occurring?” then we represent this inverse question as $s=C D F^{-1}(\alpha)$. In this illustration, $10=C D F^{-1}(0.309)$. The inverse of the right-hand pdf graph appears to be not unique. However, it might be. If the question is to determine the s-value for which the $p d f=0.113$, there is only one value, $s=10$. It appears that an $s$-value of about $13.5$ could have such a CDF value, but the count must be an integer. The $p d f$ of $S=13$ is $0.126$, and the $p d f$ of $S=14$ is $0.104$.

Although one could ask, “What count value, or lower, has a $30 \%$ chance of occurring?” it is impossible to match the $30 \% C D F=0.3 \overline{000}$ value. $S \leq 9$ has a $C D F$ of about $0.196$ which does not include the target $0.3 \overline{000} . S \leq 10$ has a CDF of about $0.309$ which does match. $S=10$ is the lowest value that includes the target $C D F$. One convention is to report the minimum count that includes the target $C D F$ value.

## 统计代写|工程统计作业代写Engineering Statistics代考|Propagating Distributions with Variable Transformations

Often, we know the distribution on $x$-values and have a model that transforms $x$ to $y$. For instance, $y=\operatorname{Ln}(x)$. The question is, “What is the distribution of $y$ ?”

Figure $3.18$ reveals the case of $y=a+b x^{3}$ when the distribution on $x$ (on the abscissa) is normal.

Note: For the range of $x$-values shown, the function is strictly monotonic, positive definite. As $x$ increases, $y$ increases for all values of $x$. There are no places in the $x$-range where either 1) the derivative is negative or 2) zero (there are no flat spots in the function).

The inset sketches indicate the $p d f$ (dashed line) and CDF of $x$ and $y$, about a nominal value of $x_{0}=2.5$ and the corresponding $y_{0}=a+b x_{0}{ }^{3}$. Note that the $p d f$ of $x$ is symmetric, and that of $y$ is skewed.

The CDF of $x$ indicates the probability that $x$ could have a lower value. For any $x$ there is a corresponding $y$, and since the function is strictly monotonic, the probability of a lower $y$-value is the same as the probability of a lower $y$-value. Then
$$\operatorname{CDF}(y=f(x))=\operatorname{CDF}(x)$$
Between any two corresponding points $x_{1}$ and $x_{2}$ separated by $\Delta x=x_{2}-x_{1}$, there are the two corresponding points $y_{1}=f\left(x_{1}\right)$ and $y_{2}=f\left(x_{2}\right)$ separated by $\Delta y=f\left(x_{2}\right)-f\left(x_{1}\right) \cong \frac{d y}{d x} \Delta x$ for small $\Delta x$ values (meaning that $\frac{d y}{d x}$ is relatively unchanged over the $\Delta x$ interval). Since $\operatorname{CDF}\left(y_{2}\right)=\operatorname{CDF}\left(x_{2}\right)$ and $\operatorname{CDF}\left(y_{1}\right)=\operatorname{CDF}\left(x_{1}\right)$, the difference is also equal, and by definition:
$$\int_{y_{1}}^{y_{2}} p d f(y) d y=\int_{x_{1}}^{x_{2}} p d f(x) d x$$
For small $\Delta x$ intervals, the integral can be approximated by the trapezoid rule of integration, and in the limit of very small $\Delta x$,

To obtain the $C D F(y)$ numerically integrate the $p d f(y)$. Using the trapezoid rule of integration, with y sorted in ascending order.
$$\operatorname{CDF}\left(y_{i+1}\right)=\operatorname{CDF}\left(y_{i}\right)+\frac{1}{2}\left[p d f\left(y_{i+1}\right)+p d f\left(y_{i}\right)\right]\left(y_{i+1}-y_{i}\right)$$
Initialize $C D F\left(y_{\text {very low }}\right)=0$.

## 统计代写|工程统计作业代写Engineering Statistics代考|For Continuum-Valued Variables

CDF(x) 与pdF(X)是
CDF⁡(X)=∫X哑剧 XpdF(X)dX

## 统计代写|工程统计作业代写Engineering Statistics代考|For Discrete-Valued Variables

CDF⁡(X)=∑X最大 XpdF(X)

## 统计代写|工程统计作业代写Engineering Statistics代考|Propagating Distributions with Variable Transformations

CDF 的X表示概率X可能有较低的价值。对于任何X有对应的是，并且由于该函数是严格单调的，因此较低的概率是-值与较低的概率相同是-价值。然后
CDF⁡(是=F(X))=CDF⁡(X)

∫是1是2pdF(是)d是=∫X1X2pdF(X)dX

CDF⁡(是一世+1)=CDF⁡(是一世)+12[pdF(是一世+1)+pdF(是一世)](是一世+1−是一世)

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## MATLAB代写

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