Probability question: Joint Probability Density Functions

Question

Two birds have landed on a power line that spans the 100' distance between utility poles.

a) What is the average distance between the birds?

b)The line runs north and south. Another bird lands on the line. What is the expected position of the north-most bird from the south-most pole?

Answer 1

For both a) and b) we can consider a related problem with the length of the power line being 1'.  Then the answer to the stated problem will be 100 x the result for the related problem.

 

For for the related problem  part a)

The normalized joint distribution function p( x1, x2)  = 1.   for 0≤ x1 ≤ 1 and  0 ≤ x2 ≤ 1

 We want  ∫∫ p(x1, x2) abs(x1 - x2)  dx2  dx1

  This can be evaluated by splitting the range of integration on the inner (x2) integral into two parts:

   0  to x1   and x1 to 1.    This yields an integrand for the outer (x1) integral which is essentially x1^2.   Integrating this over x1 from 0 to 1 gives 1/3. 

 

  So the answer to the given problem is 100/3

 

For the related problem part b)

 

  The cumulative distribution function for the maximum distance, xm, is P(xm) =xm^3.

   Thus the probability distribution function  is the derivative of this"  p(xm) = 3 xm^2

   The expected value of xm  is ∫ xm p(xm) = 3/4.

So the answer to given problem is  (3/4) 100 -  75' north of the south-most pole.