统计代写|统计推断代写Statistical inference代考|MAST90100

2022年9月29日

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统计代写|统计推断代写Statistical inference代考|Distribution and mass/density for g(X)

Suppose that $X$ is a random variable defined on $(\Omega, \mathcal{F}, \mathrm{P})$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ is a well-behaved function. We would like to derive an expression for the cumulative distribution function of $Y$, where $Y=g(X)$. Some care is required here. We define $g$ to be a function, so for every real number input there is a single real number output. However, $g^{-1}$ is not necessarily a function, so a single input may have multiple outputs. To illustrate, let $g(x)=x^2$, then $g^{-1}$ corresponds to taking the square root, an operation that typically has two real outputs; for example, $g^{-1}(4)={-2,2}$. So, in general,
$$\mathrm{P}(Y \leq y)=\mathrm{P}(g(X) \leq y) \neq \mathrm{P}\left(X \leq g^{-1}(y)\right) .$$
Our first step in deriving an expression for the distribution function of $Y$ is to consider the probability that $Y$ takes values in a subset of $\mathbb{R}$. We will use the idea of the inverse image of a set.
Definition 3.6.1 (Inverse image)
If $g: \mathbb{R} \rightarrow \mathbb{R}$ is a function and $B$ is a subset of real numbers, then the inverse image of $B$ under $g$ is the set of real numbers whose images under $g$ lie in $B$, that is, for all $B \subseteq \mathbb{R}$ we define the inverse image of $B$ under $g$ as
$$g^{-1}(B)={x \in \mathbb{R}: g(x) \in B} .$$
Then for any well-behaved $B \subseteq \mathbb{R}$,
\begin{aligned} \mathrm{P}(Y \in B) &=\mathrm{P}(g(X) \in B)=\mathrm{P}({\omega \in \Omega: g(X(\omega)) \in B}) \ &=\mathrm{P}\left(\left{\omega \in \Omega: X(\omega) \in g^{-1}(B)\right}\right)=\mathrm{P}\left(X \in g^{-1}(B)\right) . \end{aligned}
Stated loosely, the probability that $g(X)$ is in $B$ is equal to the probability that $X$ is in the inverse image of $B$. The cumulative distribution function of $Y$ is then
\begin{aligned} F_Y(y) &=\mathrm{P}(Y \leq y)=\mathrm{P}(Y \in(-\infty, y])=\mathrm{P}(g(X) \in(-\infty, y]) \ &=\mathrm{P}\left(X \in g^{-1}((-\infty, y])\right) \ &= \begin{cases}\sum_{{x: g(x) \leq y}} f_X(x) & \text { if } X \text { is discrete, } \ \int_{{x: g(x) \leq y}} f_X(x) d x & \text { if } X \text { is continuous. }\end{cases} \end{aligned}
In the discrete case, we can use similar reasoning to provide an expression for the mass function,
$$f_Y(y)=\mathrm{P}(Y=y)=\mathrm{P}(g(X)=y)=\mathrm{P}\left(X \in g^{-1}(y)\right)=\sum_{{x: g(x)=y}} f_X(x) .$$

统计代写|统计推断代写Statistical inference代考|Sequences of random variables and convergenec

Suppose that $x_1, x_2, \ldots$ is a sequence of real numbers. We denote this sequence $\left{x_n\right}$. The definition of convergence for a sequence of real numbers is well established.
Definition 3.7.1 (Convergence of a real sequence)
Let $\left{x_n\right}$ be a sequence of real numbers and let $x$ be a real number. We say that $x_n$ converges to $x$ if and only if, for every $\varepsilon>0$, we can find an integer $N$ such that $\left|x_n-x\right|<\varepsilon$ for all $n>N$. Under these conditions, we write $x_n \rightarrow x$ as $n \rightarrow \infty$.
This definition is based on an intuitively appealing idea (although in the formal statement given above, this might not be obvious). If we take any interval around $x$, say $[x-\varepsilon, x+\varepsilon]$, we can find a point, say $N$, beyond which all elements of the sequence fall in the interval. This is true for an arbitrarily small interval.

Now consider a sequence of random variables $\left{X_n\right}$ and a random variable $X$. We want to know what it means for $\left{X_n\right}$ to converge to $X$. Using Definition 3.7.1 is not possible; since $\left|X_n-X\right|$ is a random variable, direct comparison with the real number $\varepsilon$ is not meaningful. In fact, for a random variable there are many different forms of convergence. We define four distinct modes of convergence for a sequence of random variables.
Definition 3.7.2 (Types of convergence)
Let $\left{X_n\right}$ be a sequence of random variables and let $X$ be a random variable.
i. Convergence in distribution: $\left{X_n\right}$ converges in distribution to $X$ if
$$\mathrm{P}\left(X_n \leq x\right) \rightarrow \mathrm{P}(X \leq x) \text { as } n \rightarrow \infty,$$
for all $x$ at which the cumulative distribution function is continuous. This is de-noted by $X_n \stackrel{d}{\rightarrow} X$. This could also be written as $F_{X_n}(x) \rightarrow F_X(x)$. Convergence in distribution is sometimes referred to as convergence in law.
ii. Convergence in probability: $\left{X_n\right}$ converges in probability to $X$ if, for any $\varepsilon>0$
$$\mathrm{P}\left(\left|X_n-X\right|<\varepsilon\right) \rightarrow 1 \text { as } n \rightarrow \infty .$$ This is denoted $X_n \stackrel{p}{\rightarrow} X$. An alternative statement of convergence in probability is that $\mathrm{P}\left(\left|X_n-X\right|>\varepsilon\right) \rightarrow 0$ as $n \rightarrow \infty$. Convergence in probability is sometimes referred to as convergence in measure.
iii. Convergence almost surely: $\left{X_n\right}$ converges to $X$ almost surely if, for any $\varepsilon>0$,
$$\mathrm{P}\left(\lim {n \rightarrow \infty}\left|X_n-X\right|<\varepsilon\right)=1 .$$ This is denoted $X_n \stackrel{a{. s .}}{\longrightarrow} X$. An alternative statement of almost-sure convergence is that, if we define $A=\left{\omega \in \Omega: X_n(\omega) \rightarrow X(\omega)\right.$ as $\left.n \rightarrow \infty\right}$, then $\mathrm{P}(A)=1$. Almost-sure convergence is sometimes referred to as convergence with probability 1.
iv. Convergence in mean square: $\left{X_n\right}$ converges to $X$ in mean square if
$$\mathbb{E}\left[\left(X_n-X\right)^2\right] \rightarrow 0 \text { as } n \rightarrow \infty$$
This is denoted $X_n \stackrel{m \cdot s .}{\longrightarrow} X$.

统计推断代考

统计代写|统计推断代写统计推断代考| g(X)

$$\mathrm{P}(Y \leq y)=\mathrm{P}(g(X) \leq y) \neq \mathrm{P}\left(X \leq g^{-1}(y)\right) .$$的分布函数表达式的第一步 $Y$ 是要考虑概率吗 $Y$ 的子集中的值 $\mathbb{R}$。我们将使用集合的逆像的概念。

$g: \mathbb{R} \rightarrow \mathbb{R}$ 是一个函数 $B$ 是实数的子集，那么反像呢 $B$ 在 $g$ 这组实数的图像在谁的下面 $g$ 睡懒觉 $B$，也就是说，对所有人来说 $B \subseteq \mathbb{R}$ 我们定义的逆像 $B$ 在 $g$ as
$$g^{-1}(B)={x \in \mathbb{R}: g(x) \in B} .$$

\begin{aligned} \mathrm{P}(Y \in B) &=\mathrm{P}(g(X) \in B)=\mathrm{P}({\omega \in \Omega: g(X(\omega)) \in B}) \ &=\mathrm{P}\left(\left{\omega \in \Omega: X(\omega) \in g^{-1}(B)\right}\right)=\mathrm{P}\left(X \in g^{-1}(B)\right) . \end{aligned}

\begin{aligned} F_Y(y) &=\mathrm{P}(Y \leq y)=\mathrm{P}(Y \in(-\infty, y])=\mathrm{P}(g(X) \in(-\infty, y]) \ &=\mathrm{P}\left(X \in g^{-1}((-\infty, y])\right) \ &= \begin{cases}\sum_{{x: g(x) \leq y}} f_X(x) & \text { if } X \text { is discrete, } \ \int_{{x: g(x) \leq y}} f_X(x) d x & \text { if } X \text { is continuous. }\end{cases} \end{aligned}在离散的情况下，我们可以使用类似的推理来提供质量函数

统计代写|统计推断代写统计推断代考|随机变量的序列和收敛

Let $\left{X_n\right}$ 是一个随机变量序列，让 $X$ 是一个随机变量。
i。分布的趋同: $\left{X_n\right}$ 在分布上收敛于 $X$ if
$$\mathrm{P}\left(X_n \leq x\right) \rightarrow \mathrm{P}(X \leq x) \text { as } n \rightarrow \infty,$$
for all $x$ 其中累积分布函数是连续的。这被描述为 $X_n \stackrel{d}{\rightarrow} X$。这也可以写成 $F_{X_n}(x) \rightarrow F_X(x)$。分布上的收敛有时被称为规律上的收敛。概率上的收敛: $\left{X_n\right}$ 在概率上收敛于 $X$ 如果，对于任何 $\varepsilon>0$
$$\mathrm{P}\left(\left|X_n-X\right|<\varepsilon\right) \rightarrow 1 \text { as } n \rightarrow \infty .$$ 这是表示 $X_n \stackrel{p}{\rightarrow} X$。关于概率收敛的另一种说法是 $\mathrm{P}\left(\left|X_n-X\right|>\varepsilon\right) \rightarrow 0$ 作为 $n \rightarrow \infty$。概率上的收敛有时被称为度量上的收敛。几乎可以肯定的是: $\left{X_n\right}$ 收敛到 $X$ 几乎可以肯定，如果，任何 $\varepsilon>0$，
$$\mathrm{P}\left(\lim {n \rightarrow \infty}\left|X_n-X\right|<\varepsilon\right)=1 .$$ 这是表示 $X_n \stackrel{a{. s .}}{\longrightarrow} X$。另一种几乎肯定收敛的说法是，如果我们定义 $A=\left{\omega \in \Omega: X_n(\omega) \rightarrow X(\omega)\right.$ 作为 $\left.n \rightarrow \infty\right}$，那么 $\mathrm{P}(A)=1$。几乎肯定收敛有时被称为概率为1的收敛。均方收敛: $\left{X_n\right}$ 收敛到 $X$ 均方如果
$$\mathbb{E}\left[\left(X_n-X\right)^2\right] \rightarrow 0 \text { as } n \rightarrow \infty$$

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