统计代写|随机信号处理作业代写Statistical Signal Processing代考|Random Processes

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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|Random Processes

It is straightforward conceptually to go from one random variable to $k$ random variables constituting a $k$-dimensional random vector. It is perhaps a greater leap to extend the idea to a random process. The idea is at least easy to state, but it will take more work to provide examples and the mathematical details will prove more complicated. A random process is a sequence of random variables $\left{X_{n} ; n=0,1, \ldots\right}$ defined on a common experiment. It can be thought of as an infinite dimensional random vector. To be more accurate, this is an example of a discrete-time, one-sided random process. It is called “discrete-time” because the index $n$ which corresponds to time takes on discrete values (here the nonnegative integers) and it is called “one-sided” because only nonnegative times are allowed. A discrete-time random process is also called a time series in the statistics literature and it is often denoted as ${X(n) n=0,1, \ldots}$ and is sometimes denoted by ${X[n]}$ in the digital signal processing literature. Two questions might oocur to the reader: how does one construct an infinite family of random variables on a single experiment? How can one provide a direct development of a random process as accomplished for random variables and vectors? The direct development might appear hopeless since infinite dimensional vectors are involved.

The first problem is reasonably easy to handle by example. Consider the usual uniform pdf experiment. Rename the random variables $Y$ and $W$ as $X_{0}$ and $X_{1}$, respectively. Consider the following definition of an infinite family of random variables $X_{n}:[0,1) \rightarrow{0,1}$ for $n=0,1, \ldots$. Every $r \in[0,1)$ can be expanded as a binary expansion of the form
$$r=\sum_{n=0}^{\infty} b_{n}(r) 2^{-n-1}$$
This simply replaces the usual decimal representation by a binary representation. For example, $1 / 4$ is $.25$ in decimal and .01 or .010000… in binary. $1 / 2$ is .5 in decimal and yields the binary sequence .1000…, $1 / 4$ is $.25$ in decimal and yields the binary sequence .0100…, $3 / 4$ is .75 in decimal and $.11000 \ldots .$ and $1 / 3$ is $.3333 . . .$ in decimal and $.010101 \ldots$ in binary.

Define the random process by $X_{n}(r)=b_{n}(r)$, that is, the $n$th term in the binary expansion of $r$. When $n=0,1$ this reduces to the specific $X_{0}$ and $X_{1}$ already considered.

统计代写|随机信号处理作业代写Statistical Signal Processing代考|Random Variables

We now develop the promised precise definition of a random variable. As you might guess, a technical condition for random variables is required because of certain subtle pathological problems that have to do with the ability to determine probabilities for the random variable. To arrive at the precise definition, we start with the informal definition of a random variable that we have already given and then show the inevitable difficulty that results without the technical condition. We have informally defined a

random variable as being a function on a sample space. Suppose we have a probability space $(\Omega, \mathcal{F}, P)$. Let $f: \Omega \rightarrow \Re$ be a function mapping the same space into the real line so that $f$ is a candidate for a random variable. Since the selection of the original sample point $\omega$ is random, that is, governed by a probability measure, so should be the output of our measurement of random variable $f(\omega)$. That is, we should be able to find the probability of an “output event” such as the event “the outcome of the random variable $f$ was between $a$ and $b, “$ that is, the event $F \subset \Re$ given by $F=(a, b)$. Observe that there are two different kinds of events being considered here:

1. output events or members of the event space of the range or range space of the random variable, that is, events consisting of subsets of possible output values of the random variable; and
2. input events or $\Omega$ events, events in the original sample space of the original probability space.

Can we find the probability of this output event? That is, can we make mathematieal senee out of the quantity “the probability that $f$ areumee a value in an event $F \subset \Re$ ?? On reflection it seems clear that we can. The probability that $f$ assumes a value in some set of values must be the probability of all values in the original sample space that result in a value of $f$ in the given set. We will make this concept more precise shortly. To save writing we will abbreviate such English statements to the form $\operatorname{Pr}(f \in F)$, or $\operatorname{Pr}(F)$, that is, when the notation $\operatorname{Pr}(F)$ is encountered it should be interpreted as shorthand for the English statement for “the probability of an event $F^{” \prime}$ or “the probability that the event $F$ will occur” and not as a precise mathematical quantity.

统计代写|随机信号处理作业代写Statistical Signal Processing代考|Distributions of Random Variables

Suppose we have a probability space $(\Omega, \mathcal{F}, P)$ with a random variable, $X$, defined on the space. The random variable $X$ takes values on its range space which is some subset $A$ of $\Re$ (possibly $A=\Re$ ). The range space $A$ of a random variable is often called the alphabet of the random variable. As we have seen, since $X$ is a random variable, we know that all subsets of $\Omega$ of the form $X^{-1}(F)={\omega: X(\omega) \in F}$, with $F \in B(A)$, must be members of $\mathcal{F}$ by definition. Thus the set function $P_{X}$ defined by
$$P_{X}(F)=P\left(X^{-1}(F)\right)=P({\omega: X(\omega) \in F}) ; F \in \mathcal{B}(A)$$
is well defined and assigns probabilities to output events involving the random variable in terms of the original probability of input events in the orig-

inal experiment. The three written forms in equation (3.22) are all read as $\operatorname{Pr}(X \in F)$ or “the probability that the random variable $X$ takes on a value in $F .$ Furthermore, since inverse images preserve all set-theoretic operations (see problem A.12), $P_{X}$ satisfies the axioms of probability as a probability measure on $(A, \mathcal{B}(A))-$ it is nonnegative, $P_{X}(A)=1$, and it is countably additive. Thus $P_{X}$ is a probability measure on the measurable space $(A, \mathcal{B}(A))$. Therefore, given a probability space and a random variable $X$, we have constructed a new probability space $\left(A, \mathcal{B}(A), P_{X}\right)$ where the events describe outcomes of the random variable. The probability measure $P_{X}$ is called the distribution of $X$ (as opposed to a “cumulative distribution function” of $X$ to be introduced later).

If two random variables have the same distribution, then they are said to be equivalent since they have the same probabilistic description, whether or not they are defined on the same underlying space or have the same functional form (see problem 3.22).

A substantial part of the application of probability theory to practical probblems is devoted to determining the distributions of random variables, perfurmaing the “ealeulus of prubabsility.” Ons bugina with a probubility space. A random variable is defined on that space. The distribution of the random variable is then derived, and this results in a new probability space. This topic is called variously “derived distributions” or “transformations of random variables” and is often developed in the literature as a sequence of apparently unrelated subjects. When the points in the original sample space can be interpreted as “signals,” then such problems can be viewed as “signal processing” and derived distribution problems are fundamental to the analysis of statistical signal processing systems. We shall emphasize that all such examples are just applications of the basic inverse image formula (3.22) and form a unified whole. In fact, this formula, with its vector analog, is one of the most important in applications of probability theory. Its specialization to discrete input spaces using sums and to continuous input spaces using integrals will be seen and used often throughout this book.

统计代写|随机信号处理作业代写Statistical Signal Processing代考|Random Processes

r=∑n=0∞bn(r)2−n−1

统计代写|随机信号处理作业代写Statistical Signal Processing代考|Random Variables

1. 输出事件或随机变量的范围或范围空间的事件空间的成员，即由随机变量的可能输出值的子集组成的事件；和
2. 输入事件或Ω事件，原始概率空间的原始样本空间中的事件。

有限元方法代写

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MATLAB代写

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