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Annotation 7609073929484

随机过程的期望值

For any random process \(X_{t}\), the Expected Value is \(\mathbf{E}[X_{t}]=\int_{\Omega}X_{t}(\alpha)P(d\alpha)=\int_\mathbb{R}x\mu_{t}(dx)\)

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Flashcard 7609075764492

Question

For any random process \(X_{t}\), the Expected Value is [...]

Answer

\(\mathbf{E}[X_{t}]=\int_{\Omega}X_{t}(\alpha)P(d\alpha)=\int_\mathbb{R}x\mu_{t}(dx)\)

status	not learned	measured difficulty	37% [default]	last interval [days]
repetition number in this series	0	memorised on		scheduled repetition
scheduled repetition interval		last repetition or drill

随机过程的期望值
For any random process \(X_{t}\), the Expected Value is \(\mathbf{E}[X_{t}]=\int_{\Omega}X_{t}(\alpha)P(d\alpha)=\int_\mathbb{R}x\mu_{t}(dx)\)

Annotation 7609077337356

随机过程的方差

For any random process \(X_{t}\), the Varience is \(\operatorname{Var}[X_{t}]=\mathbf{E}[(X_{t}-\mathbf{E}[X_{t}])^{2}]\)

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Flashcard 7609079172364

Question

For any random process \(X_{t}\), the Varience is [...]

Answer

\(\operatorname{Var}[X_{t}]=\mathbf{E}[(X_{t}-\mathbf{E}[X_{t}])^{2}]\)

status	not learned	measured difficulty	37% [default]	last interval [days]
repetition number in this series	0	memorised on		scheduled repetition
scheduled repetition interval		last repetition or drill

随机过程的方差
For any random process \(X_{t}\), the Varience is \(\operatorname{Var}[X_{t}]=\mathbf{E}[(X_{t}-\mathbf{E}[X_{t}])^{2}]\)

Annotation 7609080745228

随机过程的自相关函数

For any random process \(X_{t}\), the Autocorrelation at time instants \(t_{1}\) and \(t_{2}\) is:

\(\displaystyle\mathbf{E}[X_{t_{1}}X_{t_{2}}]=\int_{\Omega}X_{t_{1}}(\alpha)X_{t_{2}}(\alpha)P(d\alpha)=\int_{\mathbb{R}^{2}}x_{1}x_{2}\mu_{t_{1}t_{2}}(dx_{1}\times dx_{2})\)

where \(\mu_{t_{1}t_{2}}(dx_{1}\times dx_{2})\) is a joint probability distribution.

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Flashcard 7609082580236

Question

For any random process \(X_{t}\), the Autocorrelation at time instants \(t_{1}\) and \(t_{2}\) is: [...]

Answer

\(\displaystyle\mathbf{E}[X_{t_{1}}X_{t_{2}}]=\int_{\Omega}X_{t_{1}}(\alpha)X_{t_{2}}(\alpha)P(d\alpha)=\int_{\mathbb{R}^{2}}x_{1}x_{2}\mu_{t_{1}t_{2}}(dx_{1}\times dx_{2})\)

where \(\mu_{t_{1}t_{2}}(dx_{1}\times dx_{2})\) is a joint probability distribution.

status	not learned	measured difficulty	37% [default]	last interval [days]
repetition number in this series	0	memorised on		scheduled repetition
scheduled repetition interval		last repetition or drill

随机过程的自相关函数
For any random process \(X_{t}\), the Autocorrelation at time instants \(t_{1}\) and \(t_{2}\) is: \(\displaystyle\mathbf{E}[X_{t_{1}}X_{t_{2}}]=\int_{\Omega}X_{t_{1}}(\alpha)X_{t_{2}}(\alpha)P(d\alpha)=\int_{\mathbb{R}^{2}}x_{1}x_{2}\mu_{t_{1}t_{2}}(dx_{1}\times dx_{2})\) where \(\mu_{t_{1}t_{2}}(dx_{1}\times dx_{2})\) is a joint probability distribution.

Annotation 7609091493132

稳态信号条件

The process \(X_{t}\) is said to be stationary if for any \(t_{1},t_{2},t_{3}, . . .\) and \(τ\), \(\mu_{t_{1},t_{2},t_{3},\cdots}=\mu_{t_{1}+\tau,t_{2}+\tau,t_{3}+\tau,\cdots}\)

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Flashcard 7609093590284

Question

The process \(X_{t}\) is said to be stationary if [...]

Answer

for any \(t_{1},t_{2},t_{3}, . . .\) and \(τ\), \(\mu_{t_{1},t_{2},t_{3},\cdots}=\mu_{t_{1}+\tau,t_{2}+\tau,t_{3}+\tau,\cdots}\)

status	not learned	measured difficulty	37% [default]	last interval [days]
repetition number in this series	0	memorised on		scheduled repetition
scheduled repetition interval		last repetition or drill

稳态信号条件
The process \(X_{t}\) is said to be stationary if for any \(t_{1},t_{2},t_{3}, . . .\) and \(τ\), \(\mu_{t_{1},t_{2},t_{3},\cdots}=\mu_{t_{1}+\tau,t_{2}+\tau,t_{3}+\tau,\cdots}\)

Annotation 7609095163148

广义稳态信号条件

The process \(X_{t}\) is said to be wide sense stationary (WSS) if \(\begin{align}\mathbf{E}[X_{t}]&=\text{constant}\\\mathbf{E}[X_{t_{1}}X_{t_{2}}]&=\text{depends only on }|t_{1}-t_{2}|\end{align}\)

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Flashcard 7609096998156

Question

The process \(X_{t}\) is said to be wide sense stationary (WSS) if [...]

Answer

\(\begin{align}\mathbf{E}[X_{t}]&=\text{constant}\\\mathbf{E}[X_{t_{1}}X_{t_{2}}]&=\text{depends only on }|t_{1}-t_{2}|\end{align}\)

status	not learned	measured difficulty	37% [default]	last interval [days]
repetition number in this series	0	memorised on		scheduled repetition
scheduled repetition interval		last repetition or drill

广义稳态信号条件
The process \(X_{t}\) is said to be wide sense stationary (WSS) if \(\begin{align}\mathbf{E}[X_{t}]&=\text{constant}\\\mathbf{E}[X_{t_{1}}X_{t_{2}}]&=\text{depends only on }|t_{1}-t_{2}|\end{align}\)

Annotation 7609098571020

稳态信号或广义稳态信号的自相关函数定义

For a stationary or WSS process, the autocorrelation function is defined as \(r_{XX}(\tau)=\mathbf{E}[X_{t}X_{t+\tau}]\)

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Flashcard 7609099881740

Question

For a stationary or WSS process, the autocorrelation function is defined as [...]

Answer

\(r_{XX}(\tau)=\mathbf{E}[X_{t}X_{t+\tau}]\)

status	not learned	measured difficulty	37% [default]	last interval [days]
repetition number in this series	0	memorised on		scheduled repetition
scheduled repetition interval		last repetition or drill

稳态信号或广义稳态信号的自相关函数定义
For a stationary or WSS process, the autocorrelation function is defined as \(r_{XX}(\tau)=\mathbf{E}[X_{t}X_{t+\tau}]\)

Annotation 7609107221772

伯努利分布的定义

伯努利分布的定义：\(\mu[p]=p \delta_0+(1-p) \delta_1\)

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Flashcard 7609109056780

Question

伯努利分布的定义：[...]

Answer

\(\mu[p]=p \delta_0+(1-p) \delta_1\)

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伯努利分布的定义
伯努利分布的定义：\(\mu[p]=p \delta_0+(1-p) \delta_1\)

Annotation 7609110629644

伯努利分布的期望

伯努利分布的期望：\(\mathrm{E}[X] =p\)

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Annotation 7609114037516

伯努利分布的方差

伯努利分布的方差：\(\operatorname{Var}[X] =p(1-p)\)

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Annotation 7609117445388

二项分布的定义

二项分布的定义：\(\displaystyle\mu[p]=\sum_{k=0}^n\left(\begin{array}{l}n \\k\end{array}\right) p^k(1-p)^{n-k} \delta_k\)

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Annotation 7609124261132

二项分布的期望

二项分布的期望：\(\mathrm{E}[X] =np\)

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Annotation 7609127669004

二项分布的方差

二项分布的方差：\(\operatorname{Var}[X] =np(1-p)\)

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Edited, memorised or added to reading queue

on 02-Jan-2024 (Tue)