Revisions to Conditional Expectation of Product of Normals given a Linear Combination

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Zhanxiong

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edited Mar 31, 2023 at 4:18

Zhanxiong

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I am tasked with solving a question for a qualifying exam, but I am a little lost about this question.

Let $\eta$ and $\xi$ be two independent standard Gaussian random variables. Find $\mathbb{E}(\xi\eta \mid \xi - 2\eta)$.

My attempt:

My main idea is to show that $Cov(\xi-2\eta,\xi+2\eta)=0$$\operatorname{Cov}(\xi-2\eta,\xi+2\eta)=0$, making the two independent since they are Gaussian. However, computing above, you get $Cov(\xi-2\eta,\xi+2\eta)=Var(\xi)-4Var(\eta)\neq 0$$\operatorname{Cov}(\xi-2\eta,\xi+2\eta)=\operatorname{Var}(\xi)-4\operatorname{Var}(\eta)\neq 0$.Alternatively, my thoughts were showing $Cov(\xi-\eta,\xi+\eta)=0$$\operatorname{Cov}(\xi-\eta,\xi+\eta)=0$, which is true. But is $\xi-\eta \perp \xi+\eta \implies \xi-2\eta \perp \xi+2\eta$? The main idea being after you show that\that $xi-2\eta \perp \xi+2\eta $$\xi-2\eta \perp \xi+2\eta $, write $\mathbb{E}(\eta\xi \mid \xi-2\eta)=\mathbb{E}(\frac{1}{8}[\xi+2\eta]^{2}+\frac{1}{8}[\xi-2\eta]^{2}\mid \xi-2\eta) $ and proceed.

Any help is appreciated.

I am tasked with solving a question for a qualifying exam, but I am a little lost about this question.

Let $\eta$ and $\xi$ be two independent standard Gaussian random variables. Find $\mathbb{E}(\xi\eta \mid \xi - 2\eta)$.

My attempt:

My main idea is to show that $Cov(\xi-2\eta,\xi+2\eta)=0$, making the two independent since they are Gaussian. However, computing above, you get $Cov(\xi-2\eta,\xi+2\eta)=Var(\xi)-4Var(\eta)\neq 0$.Alternatively, my thoughts were showing $Cov(\xi-\eta,\xi+\eta)=0$, which is true. But is $\xi-\eta \perp \xi+\eta \implies \xi-2\eta \perp \xi+2\eta$? The main idea being after you show that\ $xi-2\eta \perp \xi+2\eta $, write $\mathbb{E}(\eta\xi \mid \xi-2\eta)=\mathbb{E}(\frac{1}{8}[\xi+2\eta]^{2}+\frac{1}{8}[\xi-2\eta]^{2}\mid \xi-2\eta) $ and proceed.

Any help is appreciated.

I am tasked with solving a question for a qualifying exam, but I am a little lost about this question.

Let $\eta$ and $\xi$ be two independent standard Gaussian random variables. Find $\mathbb{E}(\xi\eta \mid \xi - 2\eta)$.

My attempt:

My main idea is to show that $\operatorname{Cov}(\xi-2\eta,\xi+2\eta)=0$, making the two independent since they are Gaussian. However, computing above, you get $\operatorname{Cov}(\xi-2\eta,\xi+2\eta)=\operatorname{Var}(\xi)-4\operatorname{Var}(\eta)\neq 0$.Alternatively, my thoughts were showing $\operatorname{Cov}(\xi-\eta,\xi+\eta)=0$, which is true. But is $\xi-\eta \perp \xi+\eta \implies \xi-2\eta \perp \xi+2\eta$? The main idea being after you show that $\xi-2\eta \perp \xi+2\eta $, write $\mathbb{E}(\eta\xi \mid \xi-2\eta)=\mathbb{E}(\frac{1}{8}[\xi+2\eta]^{2}+\frac{1}{8}[\xi-2\eta]^{2}\mid \xi-2\eta) $ and proceed.

Any help is appreciated.

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edited Mar 31, 2023 at 4:09

User1865345

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I am tasked with solving a question for a qualifying exam, but I am a little lost about this question.

Let $\eta$ and $\xi$ be two independent standard Gaussian random variables. Find $\mathbb{E}(\xi\eta \mid \xi - 2\eta)$.

Let $\eta$ andMy attempt:

My main idea is to show that $\xi$ be$Cov(\xi-2\eta,\xi+2\eta)=0$, making the two independent standardsince they are Gaussian random variables. FindHowever, computing above, you get $\mathbb{E}(\xi\eta \mid \xi - 2\eta)$$Cov(\xi-2\eta,\xi+2\eta)=Var(\xi)-4Var(\eta)\neq 0$.Alternatively, my thoughts were showing $Cov(\xi-\eta,\xi+\eta)=0$, which is true. But is $\xi-\eta \perp \xi+\eta \implies \xi-2\eta \perp \xi+2\eta$? The main idea being after you show that\ $xi-2\eta \perp \xi+2\eta $, write $\mathbb{E}(\eta\xi \mid \xi-2\eta)=\mathbb{E}(\frac{1}{8}[\xi+2\eta]^{2}+\frac{1}{8}[\xi-2\eta]^{2}\mid \xi-2\eta) $ and proceed.