Tag Archives: JCM_math230_HW11_F22

Conditional Variance

Posted on November 20, 2022 by | Comments Off on Conditional Variance

Given two random variables, we define the conditional variance of \(X\) given \(Y\) by

\[  \text{Var}(X | Y ) = E(X^2 | Y ) – (\,E(X | Y )\,)^2  \]

Show that

\[ \text{Var}(X ) = E (\text{Var}(X | Y )  ) +\text{Var}( E(X | Y ) )  \,. \]

Of course \( E(X | Y )\) is just  random variable so we have that

\[ \text{Var}( E(X | Y ) ) = E [\,E(X | Y )^2\,] – [\,E(\,E(X | Y )\,)\,]^2    \]

also is is useful to recall that \(E(E(X | Y ))= E(X )\).

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Random Walk

Posted on November 20, 2022 by | Comments Off on Random Walk

Let \(\{Z_1, Z_2, \dots, Z_n,\dots\} \) be a sequence of i.i.d random variables such that

\[  P( Z_k = a  ) = \begin{cases}   \frac14 & \text{ If } a=1 \\\frac14 & \text{ If } a=0\\\frac12 & \text{ If } a=-1 \end{cases}\]

If \(X_{n+1} = X_n + Z_n\) and \(X_0=1\) what is

  1.  \( E( X_{n+1} | X_{n}) \)?
  2.  \(  E( X_{n+1}) \)?
  3.  \( \text{Var}( X_{n+1} | X_{n}) \)?
  4. \( \text{Var}( X_{n+1} )\)?

Notice that \(X_n\) depends only \(Z_{n-1},Z_{n-2},\dots,Z_1\) and hence \(X_n\) is independent of \(Z_n\) !

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Posted in Conditional Expectation, Conditional Variance, Random Walk

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Joint density part 2

Posted on November 3, 2022 by | Comments Off on Joint density part 2

Let \(X\) and \(Y\) have joint density

\(f(x,y) = 90(y-x)^8, \quad 0<x<y<1\)

  1. State the conditional distribution of \(X \mid Y\) and \(Y \mid X\)
  2. Are these two random variables independent?
  3. What is \( \mathbf{P}(Y  \mid X=.2 ) \) and \( \mathbf{E}(Y  \mid  X=.2) \) ?

What is \( \mathbf{P}(Y  \mid X=.2 ) \) and \( \mathbf{E}(Y  \mid  X=.2) \)

[Adapted from Pitman pg 354]

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Posted in Basic probability, Joint Distributions

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Joint, Marginal and Conditioning

Posted on November 2, 2013 by | Comments Off on Joint, Marginal and Conditioning

Let \( (X,Y)\) have joint density \(f(x,y) = e^{-y}\), for \(0<x<y\), and \(f(x,y)=0\) elsewhere.

  1. Are \(X\) and \(Y\) independent ?
  2. Compute the marginal density of \(Y\).
  3. Show that \(f_{X|Y}(x,y)=\frac1y \), for \(0<x<y\).
  4. Compute \(E(X|Y=y)\)
  5. Use the previous result to find \(E(X)\).

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Posted in Conditional Expectation, Joint Distributions

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conditional densities

Posted on March 30, 2013 by | Comments Off on conditional densities

Let \(X\) and \(Y\) have the following joint density:

\[ f(x,y)=\begin{cases}2x+2y -4xy & \text{for } 0 \leq x\leq 1 \ \text{and}\   0 \leq y \leq 1\\ 0& \text{otherwise}\end{cases}\]

  1. Find the marginal densities of \(X\) and \(Y\)
  2. find \(f_{Y|X}( y \,|\, X=\frac14)\)
  3. find \( \mathbf{E}(Y \,|\, X=\frac14)\)

[Pitman p426 # 2]

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Expectation of hierachical model

Posted on October 5, 2012 by | Comments Off on Expectation of hierachical model

Consider the following hierarchical random variable

  1. \(\lambda \sim \mbox{Geometric}(p)\)
  2. \(Y \mid \lambda \sim \mbox{Poisson}(\lambda)\)
Compute \(\mathbf{E}(Y)\).

 

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Posted in Conditional Expectation, Conditioning, Expectations, Prior/Posterior Distribution

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