### 统计代写|随机信号处理作业代写Statistical Signal Processing代考|A Single Coin Flip

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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|随机信号处理作业代写Statistical Signal Processing代考|A Single Coin Flip

The original example of a spinning wheel is continuous in that the sample space consists of a continum of possible outcomes, all points in the unit interval. Sample spaces can also be discrete, as is the case of modeling a single flip of a “fair” coin with heads labeled ” 1 ” and tails labeled ” 0 “, i.e., heads and tails are equally likely. The sample space in this example is $\Omega={0,1}$ and the probability for any event or subset of $\omega$ can be defined in a reasonable way by
$$P(F)=\sum_{r \in F} p(r)$$

or, equivalently,
$$P(F)=\sum 1_{F}(r) p(r),$$
where now $p(r)=1 / 2$ for each $r \in \Omega$. The function $p$ is called a probability mass function or $p m f$ because it is summed over points to find total probability, just as point masses are summed to find total mass in physics. Be cautioned that $P$ is defined for sets and $p$ is defined only for points in the sample space. This can be confusing when dealing with one-point or singleton sets, for example
\begin{aligned} &P({0})=p(0) \ &P({1})=p(1) \end{aligned}
This may seem too much work for such a little example, but keep in mind that the goal is a formulation that will work for far more complicated and interesting examples. This example is different from the spinning wheel in that the sample space is discrete instead of continuous and that the probabilities of events are defined by sums instead of integrals, as one should expect when doing discrete math. It is easy to verify, however, that the basic properties (2.7)-(2.9) hold in this case as well (since sums behave like integrals), which in turn implies that the simple properties (a)-(d) also hold.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|A Single Coin Flip as Signal Processing

The coin flip example can also be derived in a very different way that provides our first example of signal processing. Consider again the spinning pointer so that the sample space is $\Omega$ and the probability measure $P$ is described by (2.2) using a uniform pdf as in (2.4). Performing the experiment by spinning the pointer will yield some real number $r \in[0,1)$. Define a measurement $q$ made on this outcome by
$$q(r)= \begin{cases}1 & \text { if } r \in[0,0.5] \ 0 & \text { if } r \in(0.5,1)\end{cases}$$
This function can also be defined somewhat more economically as
$$q(r)=1_{[0,0.5]}(r)$$
This is an example of a quantizer, an operation that maps a continuots value into a discrete one. Quantization is an example of signal processing since it is a function or mapping defined on an input space, here $\Omega=[0,1)$

or $\Omega=\Re$, producing a value in some output space, here a binary space $\Omega_{g}={0,1}$. The dependence of a function on its input space or domain of definztion $\Omega 2$ and its output space or range $\Omega_{g}$, is often denoted by $q$ : $\Omega \rightarrow \Omega_{g}$. Although introduced as an example of simple signal processing, the usual name for a real-valued function defined on the sample space of a probability space is a random varzable. We shall see in the next chapter that there is an extra technical condition on functions to merit this name. but that is a detail that can be postponed.

The output space $\Omega_{g}$ can be considered as a new sample space, the space corresponding to the possible values seen by an observer of the output of the quantizer (an observer who might not have access to the original space). If we know both the probability measure on the input space and the function, then in theory we should be able to describe the probability measure that the output space inherits from the input space. Since the output space is discrete, it should be described by a pmf, say $p_{q}$. Since there are only two points, we need only find the value of $p_{q}(1)$ (or $p_{q}(0)$ since $\left.p_{q}(0)+p_{q}(1)=1\right)$. An output of 1 is seen if and only if the input sample point lies in $[0,0.5]$, so it follows easily that $p_{q}(1)=P([0,0.5])=\int_{0}^{0.5} f(r)$, dr $=0.5$, exactly the value assumed for the fair coin flip model. The pmf $p_{q}$ implies a probability measure on the output space $\Omega_{g}$ by
$$P_{q}(F)=\sum_{\omega \in F} p_{q}(\omega),$$

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Abstract vs. Concrete

It may seem strange that the axioms of probability deal with apparently abstract ideas of measures instead of corresponding physical intuition that the probability tells you something about the fraction of times specific events will occur in a sequence of trials, such as the relative frequency of a pair of dice summing to seven in a sequence of many roles, or a decision algorithm correctly detecting a single binary symbol in the presence of noise in a transmitted data file. Such real world behavior can be quantified by the idea of a relative frequency, that is, suppose the output of the $n$th trial of a sequence of trials is $x_{n}$ and we wish to know the relative frequency that $x_{n}$ takes on a particular value, say $a$. Then given an infinite sequence of trials $x=\left{x_{0}, x_{1}, x_{2}, \ldots\right}$ we could define the relative frequency of $a$ in $x$ by
$$r_{a}(x)=\lim {n \rightarrow \infty} \frac{\text { number of } k \in{0,1, \ldots, n-1} \text { for which } x{k}=a}{n}$$
For example, the relative frequency of heads in an infinite sequence of fair coin flips should be $0.5$, the relative frequency of rolling a pair of fair dice and having the sum be 7 in an infinite sequence of rolls should be $1 / 6$ since the pairs $(1,6),(6,1),(2,5),(5,2),(3,4),(4,3)$ are equally likely and form 6 of the possible 36 pairs of outcomes. Thus one might suspect that to make a rigorous theory of probability requires only a rigorous definition of probabilities as such limits and a reaping of the resulting benefits. In fact much of the history of theoretical probability consisted of attempts to acoomplish this, but unfortunately it does not work. Such limits might not exist, or they might exist and not converge to the same thing for different repetitions of the same experiment. Even when the limits do exist there is no guarantee they will behave as intuition would suggest when one tries to do calculus with probabilities, to compute probabilities of complicated events from those of simple related events. Attempts to get around these problems uniformly failed and probability was not put on a rigorous basis until the axiomatic approach was completed by Kolmogorov. The axioms do, however, capture certain intuitive aspects of relative frequencies. Relative frequencies are nonnegative, the relative frequency of the entire set of possible outcomes is one, and relative frequencies are additive in the sense that the relative frequency of the symbol $a$ or the symbol $b$ occurring. $r_{a \cup b}(x)$, is clearly $r_{a}(x)+r_{b}(x)$. Kolmogorov realized that beginning with simple axioms could lead to rigorous limiting results of the type needed, while there was no way to begin with the limiting results as part of the axioms. In fact it is the fourth axiom, a limiting version of additivity, that plays the key role in making the asymptotics work.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|A Single Coin Flip as Signal Processing

q(r)={1 如果 r∈[0,0.5] 0 如果 r∈(0.5,1)

q(r)=1[0,0.5](r)

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Abstract vs. Concrete

r一种(X)=林n→∞ 数量 ķ∈0,1,…,n−1 为此 Xķ=一种n

## 有限元方法代写

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## MATLAB代写

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