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Joint Density Poisson arrival
Let \(T_1\) and \(T_5\) be the times of the first and fifth arrivals in a Poisson arrival prices with rate \(\lambda\). Find the joint distribution of \(T_1\) and \(T_5\) .
Uniform Spacing
Let \(U_1, U_2, U_3, U_4, U_5\) be independent uniform \((0,1)\) random variables. Let \(R\) be the difference between the max and the min of the random variables. Find
- \( E( R)\)
- the joint density of the min and the max of the \(U\)’s
- \(P( R>0.5)\)
[Pitman p. 355 #14]
Joint density part 2
Let \(X\) and \(Y\) have joint density
\(f(x,y) = 90(y-x)^8, \quad 0<x<y<1\)
- State the conditional distribution of \(X \mid Y\) and \(Y \mid X\)
- Are these two random variables independent?
- 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]
Uniform distributed points given an arrival
Consider a Poisson arrival process with rate \(\lambda>0\). Let \(T\) be the time of the first arrival starting from time \(t>0\). Let \(N(s,t]\) be the number of arrivals in the time interval \((s,t]\).
Fixing an \(L>0\), define the pdf \(f(t)\) by \(f(t)dt= P(T \in dt | N(0,L]=1)\) for \(t \in (0,L]\). Show that \(f(t)\) is the pdf of a uniform random variable on the interval \([0,L]\) (independent of \(\lambda\) !).
Joint Distribution Table
Consider the following joint distribution.
If the experiment is flipping a fair coin three times, which of the following could be the random variables \(X\) and \(Y\). Select all that apply.
- \(X=\) the number of heads, \(Y=\) the number of tails.
- \(X=\) the number of tails, \(Y=\) the number of tails (i.e., \(Y=X\)).
- \(X=\) the number of heads. \(Y=3-X.\)
- \(X=\) the number of tails on the first two flips. \(Y=\) the number of tails on the last two flips.
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.
- Are \(X\) and \(Y\) independent ?
- Compute the marginal density of \(Y\).
- Show that \(f_{X|Y}(x,y)=\frac1y \), for \(0<x<y\).
- Compute \(E(X|Y=y)\)
- Use the previous result to find \(E(X)\).
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]
A Joint density example II
If \(X\) and \(Y\) have joint density function
\[f(x,y)=\frac{1}{x^2y^2} \quad; \quad x \geq 1, y\geq 1\]
- Compute the joint density fiction of \(U=XY\), \(V=X/Y\).
- What are the marginal densities of \(U\) and \(V\) ?
[Ross p295, # 54]
Joint of min and max
Let \(X_1,…,X_n \stackrel{iid}{\sim} \mbox{Exp}(\lambda) \)
Let \(V = \mbox{min}(X_1,…,X_n)\) and \(W = \mbox{max}(X_1,…,X_n)\).
What is the joint distribution of \(V,W\). Are they independent ?
Joint density part 1
Let \(X\) and \(Y\) have joint density
\(f(x,y) = 90(y-x)^8, \quad 0<x<y<1\)
- State the marginal distribution for \(X\)
- State the marginal distribution for \(Y\)
- Are these two random variables independent?
- What is \(\mathbf{P}(Y > 2X)\)
- Fill in the blanks “The density \(f(x,y)\) above is the joint density of the _________ and __________ of ten independent uniform \((0,1)\) random variables.”
[Adapted from Pitman pg 354]
Simple Joint density
Let \(X\) and \(Y\) have joint density
\[ f(x,y) = c e^{-2x -3 y} \quad (x,y>0)\]
for some \(c>0\) and \(f(x,y)=0\) otherwise. find:
- the correct value of \(c\).
- \(P( X \leq x, Y \leq y)\)
- \(f_X(x)\)
- \(f_Y(y)\)
- Are \(X\) and \(Y\) independent ? Explain your reasoning ?
Joint Distributions of Uniforms
Let \(X\) and \(Y\) be independent, each uniformly distributed on \(\{1,2,\dots,n\}\). Find:
- \(\mathbf{P}( X=Y)\)
- \(\mathbf{P}( X < Y)\)
- \(\mathbf{P}( X>Y)\)
- \(\mathbf{P}( \text{max}(X,Y)=k )\) for \(k=1,\dots,n\)
- \(\mathbf{P}( \text{min}(X,Y)=k )\) for \(k=1,\dots,n\)
- \(\mathbf{P}( X+Y=k )\) for \(k=2,\dots,2n\)
