Question

Problem Six (Continuous Joint Random Variables)

(A) Suppose Wayne invites his friend Liz to brunch at the Copley Plaza. They are coming from separate locations and agree to meet in the lobby between 11:30am and 12 noon. If they each arrive at random times which are uniformly distributed in the interval, what is the probability that the longest either one of them waits is 10 minutes?

Hint: Letting ? and ? be the time each of them arrives in minutes after 11:30am, calculate ?(|?−?|≤10) using a geometrical argument.

(B) Let the joint probability density function of RVs ? and ? be given by

f(0, y) = 0, 9: if otherwise0 <=y <=1 <=1

For just the marginal distribution of ? calculate the probability density function ??(?) and ?(?).

Hint: You will need to solve some simple integrals!

Answer 1

Given! X NU(11:30, 12:00) YNU (11:30, 12:00) Probability Density Fusetion is daar 30 OLX 230 Similarly for (x,y), Joint distributi on is, flx,y)= ( ) ?, OLX,Y 230 To find P(1X-y1 <10) =) P(-10 <X-Y210) -> p (Y-102* <4+10) Y+10 » ) { " Ane) sazdy Y-10 +10 Y-10 20 > 15Soody

: 20x10 (30) B). $ 16,4)= say , okyarli 3(-) = 3 slon) dy "J8ydy + 8x x? = 4x3, ouuel -: E(X) = 5 x 4x’dy :45 .. E (x) = 4/5 ***** -