1 The Definition of a Stochastic Process
1 随机过程的定义

Suppose that $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$$(\mathrm{\Omega },\mathcal{F},\mathbb{P})$(Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) is a probability space, and that $X:\mathrm{\Omega }\to \mathbb{R}$$X:\mathrm{\Omega }\to \mathbb{R}$X:Omega rarrRX: \Omega \rightarrow \mathbb{R} is a random variable. Recall that this means that $\mathrm{\Omega }$$\mathrm{\Omega }$Omega\Omega is a space, $\mathcal{F}$$\mathcal{F}$F\mathcal{F} is a $\sigma$$\sigma$sigma\sigma-algebra of subsets of $\mathrm{\Omega },\mathbb{P}$$\mathrm{\Omega },\mathbb{P}$Omega,P\Omega, \mathbb{P} is a countably additive, non-negative measure on $(\mathrm{\Omega },\mathcal{F})$$(\mathrm{\Omega },\mathcal{F})$(Omega,F)(\Omega, \mathcal{F}) with total mass $\mathbb{P}(\mathrm{\Omega })=1$$\mathbb{P}(\mathrm{\Omega })=1$P(Omega)=1\mathbb{P}(\Omega)=1, and $X$$X$XX is a measurable function, i.e., $X^{-1}(B)=\{\omega \in \mathrm{\Omega }:X(\omega )\in B\}\in \mathcal{F}$$X^{-1}(B)=\{\omega \in \mathrm{\Omega }:X(\omega )\in B\}\in \mathcal{F}$X^(-1)(B)={omega in Omega:X(omega)in B}inFX^{-1}(B)=\{\omega \in \Omega: X(\omega) \in B\} \in \mathcal{F} for every Borel set $B\in \mathcal{B}(\mathbb{R})$$B\in \mathcal{B}(\mathbb{R})$B inB(R)B \in \mathcal{B}(\mathbb{R}).
假设 $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$$(\mathrm{\Omega },\mathcal{F},\mathbb{P})$(Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 是一个概率空间，而 $X:\mathrm{\Omega }\to \mathbb{R}$$X:\mathrm{\Omega }\to \mathbb{R}$X:Omega rarrRX: \Omega \rightarrow \mathbb{R} 是一个随机变量。回想一下，这意味着 $\mathrm{\Omega }$$\mathrm{\Omega }$Omega\Omega 是一个空间， $\mathcal{F}$$\mathcal{F}$F\mathcal{F} 是一个 $\sigma$$\sigma$sigma\sigma 子集的代数， $\mathrm{\Omega },\mathbb{P}$$\mathrm{\Omega },\mathbb{P}$Omega,P\Omega, \mathbb{P} 是 $(\mathrm{\Omega },\mathcal{F})$$(\mathrm{\Omega },\mathcal{F})$(Omega,F)(\Omega, \mathcal{F}) 上一个可数可加的非负度量，总质量为 $\mathbb{P}(\mathrm{\Omega })=1$$\mathbb{P}(\mathrm{\Omega })=1$P(Omega)=1\mathbb{P}(\Omega)=1 ，而 $X$$X$XX 是一个可测函数，即 $X^{-1}(B)=\{\omega \in \mathrm{\Omega }:X(\omega )\in B\}\in \mathcal{F}$$X^{-1}(B)=\{\omega \in \mathrm{\Omega }:X(\omega )\in B\}\in \mathcal{F}$X^(-1)(B)={omega in Omega:X(omega)in B}inFX^{-1}(B)=\{\omega \in \Omega: X(\omega) \in B\} \in \mathcal{F} 对于每个伯尔集合 $B\in \mathcal{B}(\mathbb{R})$$B\in \mathcal{B}(\mathbb{R})$B inB(R)B \in \mathcal{B}(\mathbb{R}) 都是可测函数。

A stochastic process is simply a collection of random variables indexed by time. It will be useful to consider separately the cases of discrete time and continuous time. We will even have occasion to consider indexing the random variables by negative time. That is, a discrete time stochastic process $X=\{X_n,n=0,1,2,\dots \}$$X=\left\{X_n,n=0,1,2,\dots \right\}$X={X_(n),n=0,1,2,dots}X=\left\{X_{n}, n=0,1,2, \ldots\right\} is a countable collection of random variables indexed by the non-negative integers, and a continuous time stochastic process $X=\{X_t,0\le t<\mathrm{\infty }\}$$X=\left\{X_t,0\le t<\mathrm{\infty }\right\}$X={X_(t),0 <= t < oo}X=\left\{X_{t}, 0 \leq t<\infty\right\} is an uncountable collection of random variables indexed by the non-negative real numbers.
随机过程是以时间为索引的随机变量集合。分别考虑离散时间和连续时间的情况是有用的。我们甚至有机会考虑以负时间为索引的随机变量。也就是说，离散时间随机过程 $X=\{X_n,n=0,1,2,\dots \}$$X=\left\{X_n,n=0,1,2,\dots \right\}$X={X_(n),n=0,1,2,dots}X=\left\{X_{n}, n=0,1,2, \ldots\right\} 是以非负整数为索引的可数随机变量集合，而连续时间随机过程 $X=\{X_t,0\le t<\mathrm{\infty }\}$$X=\left\{X_t,0\le t<\mathrm{\infty }\right\}$X={X_(t),0 <= t < oo}X=\left\{X_{t}, 0 \leq t<\infty\right\} 是以非负实数为索引的不可数随机变量集合。

In general, we may consider any indexing set $I\subset \mathbb{R}$$I\subset \mathbb{R}$I subRI \subset \mathbb{R} having infinite cardinality, so that calling $X=\{X_{\alpha },\alpha \in I\}$$X=\left\{X_{\alpha },\alpha \in I\right\}$X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} a stochastic process simply means that $X_{\alpha }$$X_{\alpha }$X_(alpha)X_{\alpha} is a random variable for each $\alpha \in I$$\alpha \in I$alpha in I\alpha \in I. (If the cardinality of $I$$I$II is finite, then $X$$X$XX is not considered a stochastic process, but rather a random vector.)
一般来说，我们可以考虑任何具有无限卡片性的索引集 $I\subset \mathbb{R}$$I\subset \mathbb{R}$I subRI \subset \mathbb{R} ，因此称 $X=\{X_{\alpha },\alpha \in I\}$$X=\left\{X_{\alpha },\alpha \in I\right\}$X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} 为随机过程仅仅意味着 $X_{\alpha }$$X_{\alpha }$X_(alpha)X_{\alpha} 是每个 $\alpha \in I$$\alpha \in I$alpha in I\alpha \in I 的随机变量。(如果 $I$$I$II 的卡片数是有限的，那么 $X$$X$XX 就不能视为随机过程，而应视为随机向量）。

There are two natural questions that one might ask.
人们自然会提出两个问题。
(1) How can we construct a probability space on which a stochastic process is defined?
(1) 如何构建一个概率空间，并在其上定义随机过程？
(2) Is it possible to define a stochastic process by specifying, say, its finite dimensional distributions only?
(2) 是否有可能只指定随机过程的有限维分布来定义随机过程？

Instead of immediately addressing these (rather technical) questions, we assume the existence of an appropriate probability space, and carefully define a stochastic process on that space. In fact, we will defer answering these questions for some time.
我们不会立即讨论这些（相当技术性的）问题，而是假设存在一个适当的概率空间，并仔细定义该空间上的随机过程。事实上，我们将推迟一段时间来回答这些问题。

Definition 1.1. Suppose that $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$$(\mathrm{\Omega },\mathcal{F},\mathbb{P})$(Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) is a probability space, and that $I\subset \mathbb{R}$$I\subset \mathbb{R}$I subRI \subset \mathbb{R} is of infinite cardinality. Suppose further that for each $\alpha \in I$$\alpha \in I$alpha in I\alpha \in I, there is a random variable $X_{\alpha }:\mathrm{\Omega }\to \mathbb{R}$$X_{\alpha }:\mathrm{\Omega }\to \mathbb{R}$X_(alpha):Omega rarrRX_{\alpha}: \Omega \rightarrow \mathbb{R} defined on $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$$(\mathrm{\Omega },\mathcal{F},\mathbb{P})$(Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}). The function $X:I\times \mathrm{\Omega }\to \mathbb{R}$$X:I\times \mathrm{\Omega }\to \mathbb{R}$X:I xx Omega rarrRX: I \times \Omega \rightarrow \mathbb{R} defined by $X(\alpha ,\omega )=X_{\alpha }(\omega )$$X(\alpha ,\omega )=X_{\alpha }(\omega )$X(alpha,omega)=X_(alpha)(omega)X(\alpha, \omega)=X_{\alpha}(\omega) is called a stochastic process with indexing set $I$$I$II, and is written $X=\{X_{\alpha },\alpha \in I\}$$X=\left\{X_{\alpha },\alpha \in I\right\}$X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\}.
定义 1.1.假设 $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$$(\mathrm{\Omega },\mathcal{F},\mathbb{P})$(Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 是一个概率空间，且 $I\subset \mathbb{R}$$I\subset \mathbb{R}$I subRI \subset \mathbb{R} 的卡方数为无限。再假设每个 $\alpha \in I$$\alpha \in I$alpha in I\alpha \in I 都有一个定义在 $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$$(\mathrm{\Omega },\mathcal{F},\mathbb{P})$(Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 上的随机变量 $X_{\alpha }:\mathrm{\Omega }\to \mathbb{R}$$X_{\alpha }:\mathrm{\Omega }\to \mathbb{R}$X_(alpha):Omega rarrRX_{\alpha}: \Omega \rightarrow \mathbb{R} 。由 $X(\alpha ,\omega )=X_{\alpha }(\omega )$$X(\alpha ,\omega )=X_{\alpha }(\omega )$X(alpha,omega)=X_(alpha)(omega)X(\alpha, \omega)=X_{\alpha}(\omega) 定义的函数 $X:I\times \mathrm{\Omega }\to \mathbb{R}$$X:I\times \mathrm{\Omega }\to \mathbb{R}$X:I xx Omega rarrRX: I \times \Omega \rightarrow \mathbb{R} 称为具有索引集 $I$$I$II 的随机过程，并被写为 $X=\{X_{\alpha },\alpha \in I\}$$X=\left\{X_{\alpha },\alpha \in I\right\}$X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} 。

Remark. We will always assume that the cardinality of $I$$I$II is infinite, either countable or uncountable. If $I={\mathbb{Z}}^{+}$$I={\mathbb{Z}}^{+}$I=Z^(+)I=\mathbb{Z}^{+}, then we called $X$$X$XX a discrete time stochastic process, and if $I=[0,\mathrm{\infty })$$I=[0,\mathrm{\infty })$I=[0,oo)I=[0, \infty), then $X$$X$XX is said to be a continuous time stochastic processes.
备注我们将始终假设 $I$$I$II 的心数是无限的，可以是可数的，也可以是不可数的。如果 $I={\mathbb{Z}}^{+}$$I={\mathbb{Z}}^{+}$I=Z^(+)I=\mathbb{Z}^{+} ，那么我们称 $X$$X$XX 为离散时间随机过程；如果 $I=[0,\mathrm{\infty })$$I=[0,\mathrm{\infty })$I=[0,oo)I=[0, \infty) ，那么我们称 $X$$X$XX 为连续时间随机过程。

At first, this definition might seem a little complicated since we are regarding the stochastic process $X$$X$XX as a function of two variables defined on the product space $I\times \mathrm{\Omega }$$I\times \mathrm{\Omega }$I xx OmegaI \times \Omega. However, this is necessary since we do not always want to view the stochastic process $X$$X$XX as a collection of random variables. Sometimes, it is more advantageous to consider $X$$X$XX as the (random) function $\alpha \mapsto X(\alpha ,\omega )$$\alpha \mapsto X(\alpha ,\omega )$alpha|->X(alpha,omega)\alpha \mapsto X(\alpha, \omega) which is called the sample path (or trajectory) of $X$$X$XX at $\omega$$\omega$omega\omega (and is also written $X(\omega ))$$X(\omega ))$X(omega))X(\omega)). We will need to require $X$$X$XX as a function of $\alpha$$\alpha$alpha\alpha to have certain regularity properties such as continuity or measurability; as will be shown in Example 3.10 below, these properties do not come for free!
起初，这个定义似乎有点复杂，因为我们将随机过程 $X$$X$XX 视为定义在乘积空间 $I\times \mathrm{\Omega }$$I\times \mathrm{\Omega }$I xx OmegaI \times \Omega 上的两个变量的函数。然而，这是必要的，因为我们并不总是希望将随机过程 $X$$X$XX 视为随机变量的集合。有时，将 $X$$X$XX 视为（随机）函数 $\alpha \mapsto X(\alpha ,\omega )$$\alpha \mapsto X(\alpha ,\omega )$alpha|->X(alpha,omega)\alpha \mapsto X(\alpha, \omega) 更为有利，该函数被称为 $X$$X$XX 在 $\omega$$\omega$omega\omega 处的样本路径（或轨迹）（也可写为 $X(\omega ))$$X(\omega ))$X(omega))X(\omega)) 。我们需要要求 $X$$X$XX 作为 $\alpha$$\alpha$alpha\alpha 的函数具有某些规则性属性，例如连续性或可测性；正如下面的例 3.10 所示，这些属性并不是免费的！

Notation. A word should be said about notation. We have defined a stochastic process as a single function $X$$X$XX of two variables. That is, $X:I\times \mathrm{\Omega }\to \mathbb{R}$$X:I\times \mathrm{\Omega }\to \mathbb{R}$X:I xx Omega rarrRX: I \times \Omega \rightarrow \mathbb{R} is defined by specifying $(\alpha ,\omega )\mapsto X(\alpha ,\omega )$$(\alpha ,\omega )\mapsto X(\alpha ,\omega )$(alpha,omega)|->X(alpha,omega)(\alpha, \omega) \mapsto X(\alpha, \omega) which mimics the notation from multi-variable calculus. However, we are also viewing a stochastic process as a collection of random variables, one random variable for each $\alpha$$\alpha$alpha\alpha in the indexing set $I$$I$II. That is, if the random variable $X_{\alpha }:\mathrm{\Omega }\to \mathbb{R}$$X_{\alpha }:\mathrm{\Omega }\to \mathbb{R}$X_(alpha):Omega rarrRX_{\alpha}: \Omega \rightarrow \mathbb{R} is defined by specifying $\omega \mapsto X_{\alpha }(\omega )$$\omega \mapsto X_{\alpha }(\omega )$omega|->X_(alpha)(omega)\omega \mapsto X_{\alpha}(\omega), then the stochastic process $X$$X$XX is defined as $X(\alpha ,\omega )=X_{\alpha }(\omega )$$X(\alpha ,\omega )=X_{\alpha }(\omega )$X(alpha,omega)=X_(alpha)(omega)X(\alpha, \omega)=X_{\alpha}(\omega). In fact, we will often say for brevity that $X=\{X_{\alpha },\alpha \in I\}$$X=\left\{X_{\alpha },\alpha \in I\right\}$X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} is a stochastic process on $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$$(\mathrm{\Omega },\mathcal{F},\mathbb{P})$(Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}). Because of this identification, when there is no chance of ambiguity we will use both $X(\alpha ,\omega )$$X(\alpha ,\omega )$X(alpha,omega)X(\alpha, \omega) and $X_{\alpha }(\omega )$$X_{\alpha }(\omega )$X_(alpha)(omega)X_{\alpha}(\omega) to describe the stochastic process. If the dependence on $\omega$$\omega$omega\omega is unnecessary, we will simply write $X_{\alpha }$$X_{\alpha }$X_(alpha)X_{\alpha} or even $X(\alpha )$$X(\alpha )$X(alpha)X(\alpha). The sample path of $X$$X$XX at $\omega$$\omega$omega\omega will be written as either $\alpha \mapsto X_{\alpha }(\omega )$$\alpha \mapsto X_{\alpha }(\omega )$alpha|->X_(alpha)(omega)\alpha \mapsto X_{\alpha}(\omega) or just $X(\omega )$$X(\omega )$X(omega)X(\omega).
记号先说一下符号。我们将随机过程定义为两个变量的单一函数 $X$$X$XX 。也就是说， $X:I\times \mathrm{\Omega }\to \mathbb{R}$$X:I\times \mathrm{\Omega }\to \mathbb{R}$X:I xx Omega rarrRX: I \times \Omega \rightarrow \mathbb{R} 是通过指定 $(\alpha ,\omega )\mapsto X(\alpha ,\omega )$$(\alpha ,\omega )\mapsto X(\alpha ,\omega )$(alpha,omega)|->X(alpha,omega)(\alpha, \omega) \mapsto X(\alpha, \omega) 来定义的，这模仿了多变量微积分的符号。不过，我们也将随机过程视为随机变量的集合，索引集 $I$$I$II 中的每个 $\alpha$$\alpha$alpha\alpha 都有一个随机变量。也就是说，如果通过指定 $\omega \mapsto X_{\alpha }(\omega )$$\omega \mapsto X_{\alpha }(\omega )$omega|->X_(alpha)(omega)\omega \mapsto X_{\alpha}(\omega) 来定义随机变量 $X_{\alpha }:\mathrm{\Omega }\to \mathbb{R}$$X_{\alpha }:\mathrm{\Omega }\to \mathbb{R}$X_(alpha):Omega rarrRX_{\alpha}: \Omega \rightarrow \mathbb{R} ，那么随机过程 $X$$X$XX 的定义就是 $X(\alpha ,\omega )=X_{\alpha }(\omega )$$X(\alpha ,\omega )=X_{\alpha }(\omega )$X(alpha,omega)=X_(alpha)(omega)X(\alpha, \omega)=X_{\alpha}(\omega) 。事实上，为了简洁起见，我们经常会说 $X=\{X_{\alpha },\alpha \in I\}$$X=\left\{X_{\alpha },\alpha \in I\right\}$X={X_(alpha),alpha in I}X=\left\{X_{\alpha}, \alpha \in I\right\} 是 $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$$(\mathrm{\Omega },\mathcal{F},\mathbb{P})$(Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 上的随机过程。由于这种识别，在没有歧义的情况下，我们将同时使用 $X(\alpha ,\omega )$$X(\alpha ,\omega )$X(alpha,omega)X(\alpha, \omega) 和 $X_{\alpha }(\omega )$$X_{\alpha }(\omega )$X_(alpha)(omega)X_{\alpha}(\omega) 来描述随机过程。如果不需要依赖 $\omega$$\omega$omega\omega ，我们将简单地写成 $X_{\alpha }$$X_{\alpha }$X_(alpha)X_{\alpha} ，甚至 $X(\alpha )$$X(\alpha )$X(alpha)X(\alpha) 。 $X$$X$XX 在 $\omega$$\omega$omega\omega 处的样本路径将写成 $\alpha \mapsto X_{\alpha }(\omega )$$\alpha \mapsto X_{\alpha }(\omega )$alpha|->X_(alpha)(omega)\alpha \mapsto X_{\alpha}(\omega) 或只写成 $X(\omega )$$X(\omega )$X(omega)X(\omega) 。

Example 1.2. Perhaps the simplest example of a stochastic process is what may be termed i.i.d. noise. Suppose that $X_1,X_2,\dots$$X_1,X_2,\dots$X_(1),X_(2),dotsX_{1}, X_{2}, \ldots are independent and identically distributed random variables on a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$$(\mathrm{\Omega },\mathcal{F},\mathbb{P})$(Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) each having mean zero and variance one. The stochastic process $X:\mathbb{N}\times \mathrm{\Omega }\to \mathbb{R}$$X:\mathbb{N}\times \mathrm{\Omega }\to \mathbb{R}$X:Nxx Omega rarrRX: \mathbb{N} \times \Omega \rightarrow \mathbb{R} defined by $X(n,\omega )=X_n(\omega )$$X(n,\omega )=X_n(\omega )$X(n,omega)=X_(n)(omega)X(n, \omega)=X_{n}(\omega) is called i.i.d. noise and serves as the building block for other more complicated stochastic processes. For example, define the stochastic process $S:\mathbb{N}\times \mathrm{\Omega }\to \mathbb{R}$$S:\mathbb{N}\times \mathrm{\Omega }\to \mathbb{R}$S:Nxx Omega rarrRS: \mathbb{N} \times \Omega \rightarrow \mathbb{R} by setting
例 1.2.随机过程最简单的例子可能就是所谓的 i.i.d. 噪声。假设 $X_1,X_2,\dots$$X_1,X_2,\dots$X_(1),X_(2),dotsX_{1}, X_{2}, \ldots 是概率空间 $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$$(\mathrm{\Omega },\mathcal{F},\mathbb{P})$(Omega,F,P)(\Omega, \mathcal{F}, \mathbb{P}) 上独立且同分布的随机变量，每个变量的均值为零，方差为一。由 $X(n,\omega )=X_n(\omega )$$X(n,\omega )=X_n(\omega )$X(n,omega)=X_(n)(omega)X(n, \omega)=X_{n}(\omega) 定义的随机过程 $X:\mathbb{N}\times \mathrm{\Omega }\to \mathbb{R}$$X:\mathbb{N}\times \mathrm{\Omega }\to \mathbb{R}$X:Nxx Omega rarrRX: \mathbb{N} \times \Omega \rightarrow \mathbb{R} 称为 i.i.d. 噪声，是其他更复杂随机过程的基础。例如，定义随机过程 $S:\mathbb{N}\times \mathrm{\Omega }\to \mathbb{R}$$S:\mathbb{N}\times \mathrm{\Omega }\to \mathbb{R}$S:Nxx Omega rarrRS: \mathbb{N} \times \Omega \rightarrow \mathbb{R} 时，设置

The stochastic process $S$$S$SS is called a random walk and will be studied in greater detail later.
随机过程 $S$$S$SS 被称为随机漫步，稍后将对其进行更详细的研究。
The following section discusses some examples of continuous time stochastic processes.
下一节将讨论一些连续时间随机过程的例子。

We begin by recalling the useful fact that a linear transformation of a normal random variable is again a normal random variable.
我们首先回顾一个有用的事实：正态随机变量的线性变换也是正态随机变量。

Proposition 2.1. Suppose that $Z\sim \mathcal{N}(0,1)$$Z\sim \mathcal{N}(0,1)$Z∼N(0,1)Z \sim \mathcal{N}(0,1). If $a,b\in \mathbb{R}$$a,b\in \mathbb{R}$a,b inRa, b \in \mathbb{R} and $Y=aZ+b$$Y=aZ+b$Y=aZ+bY=a Z+b, then $Y\sim \mathcal{N}(b,a^2)$$Y\sim \mathcal{N}\left(b,a^2\right)$Y∼N(b,a^(2))Y \sim \mathcal{N}\left(b, a^{2}\right).
命题 2.1.假设 $Z\sim \mathcal{N}(0,1)$$Z\sim \mathcal{N}(0,1)$Z∼N(0,1)Z \sim \mathcal{N}(0,1) 。如果 $a,b\in \mathbb{R}$$a,b\in \mathbb{R}$a,b inRa, b \in \mathbb{R} 和 $Y=aZ+b$$Y=aZ+b$Y=aZ+bY=a Z+b ，那么 $Y\sim \mathcal{N}(b,a^2)$$Y\sim \mathcal{N}\left(b,a^2\right)$Y∼N(b,a^(2))Y \sim \mathcal{N}\left(b, a^{2}\right)