Tag Archives: JCM_math230_HW10_S15

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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A joint density example I

Posted on November 2, 2013 by | Comments Off on A joint density example I

Let \( (X,Y) \) have joint density \(f(x,y)=x e^{-x-y}\) when \(x,y>0\) and \(f(x,y)=0\) elsewhere. Are \(X\) and \(Y\) independent ?

 

[Meester ex 5.12.30]

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Posted in Independence, 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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Posted in Conditional Expectation, probability density function

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Difference between max and min

Posted on March 19, 2013 by | Comments Off on Difference between max and min

Let \(U_1,U_2,U_3,U_4,U_5\) be independent, each with uiform distribution on \((0,1)\). Let \(R\) be the distance between the max and the min of the \(U_i\)’s. Find

  1. \(\mathbf{E} R\)
  2. the joint density of the max and the min of the \(U_i\)’s.
  3. the \(\mathbf{P}(R> .5)\)

[pitman p355, #14]

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Posted in Max and Mins, Order Statistics

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Joint Density of Arrival Times

Posted on March 18, 2013 by | Leave a comment

Let \(T_1  < T_2<\cdots\) be the arrival times in a Poisson arrival process with rate \(\lambda\). What is the joint distribution of \((T_1,T_2,T_5)\) ?

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Posted in Poisson

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Conditioning and Polya’s urn

Posted on November 8, 2012 by | Comments Off on Conditioning and Polya’s urn

An urn contains 1 black and 2 white balls. One ball is drawn at random and its color noted. The ball is replaced in the urn, together with an additional ball of its color. There are now four balls in the urn. Again, one ball is drawn at random from the urn, then replaced along with an additional ball of its color. The process continues in this way.

  1. Let \(B_n\) be the number of black balls in the urn just before the \(n\)th ball is drawn. (Thus \(B_1= 1\).) For \(n \geq 1\), find \(\mathbf{E} (B_{n+1} | B_{n}) \).
  2. For \(n \geq 1\),  find \(\mathbf{E} (B_{n}) \). [Hint: Use induction based on the previous answer and the fact that \(\mathbf{E}(B_1) =1\)]
  3.   For \(n \geq 1\), what is the expected proportion of black balls in the urn just before the \(n\)th ball is drawn ?

 

[From pitman p 408, #6]

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Posted in Conditioning, Urns

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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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