Category Archives: Joint Distributions

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

  1. \( E( R)\)
  2. the joint density of the min and the max of the \(U\)’s
  3. \(P( R>0.5)\)

[Pitman p. 355 #14]

Joint density part 2

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

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

  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]

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.

  1. \(X=\) the number of heads, \(Y=\) the number of tails.
  2. \(X=\) the number of tails, \(Y=\) the number of tails (i.e., \(Y=X\)).
  3. \(X=\) the number of heads. \(Y=3-X.\)
  4. \(X=\) 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.

  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 \(E(X)\).

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

\[f(x,y)=\frac{1}{x^2y^2} \quad; \quad x \geq 1, y\geq 1\]

  1.  Compute the joint density fiction of  \(U=XY\), \(V=X/Y\).
  2. 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

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

  1. State the marginal distribution for \(X\)
  2. State the marginal distribution for \(Y\)
  3. Are these two random variables independent?
  4. What is \(\mathbf{P}(Y > 2X)\)
  5. 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:

  1. the correct value of \(c\).
  2. \(P( X \leq x, Y \leq y)\)
  3. \(f_X(x)\)
  4. \(f_Y(y)\)
  5. 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:

  1. \(\mathbf{P}( X=Y)\)
  2. \(\mathbf{P}( X < Y)\)
  3. \(\mathbf{P}( X>Y)\)
  4. \(\mathbf{P}( \text{max}(X,Y)=k )\) for \(k=1,\dots,n\)
  5. \(\mathbf{P}( \text{min}(X,Y)=k )\) for \(k=1,\dots,n\)
  6. \(\mathbf{P}( X+Y=k )\) for \(k=2,\dots,2n\)