### 统计代写|随机信号处理作业代写Statistical Signal Processing代考| Continuous Probability Spaces

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• Statistical Inference 统计推断
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• 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|Continuous Probability Spaces

Continuous spaces are handled in a manner analogous to discrete spaces, but with some fundamental differences. The primary difference is that usually probabilities are computed by integrating a density function instead of eumming a maes function. The good news is that moet formulas look the same with integrals replacing sums. The bad news is that there are some underlying theoretical issues that require consideration. The problem is that integrals are themselves limits, and limits do not always exist in the sense of converging to a finite number. Because of this, some care will be needed to clarify when the resulting probabilities are well defined.
[2.14] Let $(\Omega, \mathcal{F})=(\Re, \mathcal{B}(\Re))$, the real line together with its Borel field. Suppose that we have a real-valued function $f$ on the real line that satisfies the following properties
$$\begin{gathered} f(r) \geq 0, \text { all } r \in \Omega \ \int_{\Omega} f(r) d r=1 \end{gathered}$$
that is, the function $f(r)$ has a well-defined integral over the real line. Define the set function $P$ by
$$P(F)=\int_{F} f(r) d r=\int 1_{F}(r) f(r) d r, F \in \mathcal{B}(\Re)$$
We note that a probability space defined as a probability measure on a Borel field is an example of a Borel space.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Probabilities as Integrals

The first issue is fundamental: Does the integral of (2.56) make sense; i.e., is it well-defined for all events of interest? Suppose first that we take the common engineering approach and use Riemann integration – the form

of integration used in elementary calculus. Then the above integrals are defined at least for events $F$ that are intervals. This implies from the linearity properties of Riemann integration that the integrals are also welldefined for events $F$ that are finite unions of intervals. It is not difficult, however, to construct sets $F$ for which the indicator function $1_{F}$ is so nasty that the function $f(r) 1_{F}(r)$ does not have a Riemann integral. For example, suppose that $f(r)$ is 1 for $r \in[0,1]$ and 0 otherwise. Then the Riemann integral $\int 1_{F}(r) f(r) d r$ is not defined for the set $F$ of all irrational numbers, yet intuition should suggest that the set has probability 1 . This intuition reflects the fact that if all points are somehow equally probable, then since the unit interval contains an uncountable infinity of irrational numbers and only a countable infinity of rational numbers, then the probability of the former tet rhould bo one and that of the latter $\overline{0}$. This intuition in not reflected in the integral definition, which is not defined for either set by the Riemann approach. Thus the definition of (2.56) has a basic problem: The integral in the formula giving the probability measure of a set might not be well-defined.

A natural approach to escaping this dilemma would be to use the Riemann integral when possible, i.e., to define the probabilities of events that are finite unions of intervals, and then to obtain the probabilities of more complicated events by expressing them as a limit of finite unions of intervals, if the limit makes sense. This would hopefully give us a reasonable definition of a probability measure on a class of events much larger than the class of all finite unions of intervals. Intuitively, it should give us a probability measure of all sets that can be expressed as increasing or decreasing limits of finite unions of intervals.

This larger class is, in fact, the Borel field, but the Riemann integral has the unfortunate property that in general we cannot interchange limits and integration; that is, the limit of a sequence of integrals of converging functions may not be itself an integral of a limiting function.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Probability Density Functions

The function $f$ used in (2.54) to $(2.56)$ is called a probability density function or $p d f$ since it is a nonnegative function that is integrated to find a total mass of probability, just as a mass density function in physics is integrated to find a total mass. Like a pmf, a pdf is defined only for points in $\Omega$ and not for sets. Unlike a pmf, a pdf is not in itself the probability of anything; for example, a pdf can take on values greater than one, while a pmf cannot. Under a pdf, points frequently have probability zero, even though the pdf is nonzero. We can, however, interpret a pdf as being proportional to a probability in the following sense. For a pmf we had
$$p(x)=P({x})$$
Suppose now that the sample space is the real line and that a pdf $f$ is defined. Let $F=[x, x+\Delta x)$, where $\Delta x$ is extremely small. Then if $f$ is sufficiently smooth, the mean value theorem of calculus implies that
$$P([x, x+\Delta x))=\int_{x}^{x+\Delta x} f(\alpha) d \alpha \approx f(x) \Delta x$$
Thus if a pdf $f(x)$ is multiplied by a differential $\Delta x$, it can be interpreted as (approximately) the probability of being within $\Delta x$ of $x$.

Both probability functions, the pmf and the pdf, can be used to define and compute a probability measure: The pmf is summed over all points in the event, and the pdf is integrated over all points in the event. If the sample space is the subset of the real line, both can be used to compute expectations such as moments.

Some of the most common pdf’s are listed below. As will be seen, these are indeed valid pdf’s, that is, they satisfy (2.54) and (2.55). The pdf’s are assumed to be 0 outside of the specified domain. $b, a, \lambda>0, m$, and $\sigma>0$ are parameters in $\Re$.
The uniform pdf. Given $b>a, f(r)=1 /(b-a)$ for $r \in[a, b]$.
The exponential pdf. $f(r)=\lambda e^{-\lambda r} ; r \geq 0$.
The doubly exponential (or Laplacian) pdf. $f(r)=\frac{\lambda}{2} e^{-\lambda|r|} ; r \in$ $\Re$.

The Gaussian (or Normal) pdf. $f(r)=\left(2 \pi \sigma^{2}\right)^{-1 / 2} \exp \left(\frac{-(r-m)^{2}}{2 \sigma^{2}}\right)$; $r \in{$. Since the density is completely described by two parameters: the mean $m$ and variance $\sigma^{2}>0$, it is common to denote it by $\mathcal{N}\left(m, \sigma^{2}\right)$.
Other univariate pdf’s may be found in Appendix C.
Just as we used a pdf to construct a probability measure on the space $(\Re, \mathcal{B}(\Re)$ ), we can also use it to define a probability measure on any smaller space $(A, B(A))$, where $A$ is a subset of $\Re$.

As a technical detail we note that to ensure that the integrals all behave as expected we must also require that $A$ itself be a Borel set of $\Re$ so that it is precluded from being too nasty a set. Such probability spaces can be considered to have a sample space of either $\Re$ or $A$, as convenient. In the former case events outside of $A$ will have zero probability.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Continuous Probability Spaces

[2.14] 让(Ω,F)=(ℜ,乙(ℜ)), 实线连同它的 Borel 场。假设我们有一个实值函数F在满足下列性质的实线上
F(r)≥0, 全部 r∈Ω ∫ΩF(r)dr=1

p(X)=磷(X)

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