A stick is one foot long. You break it at a point X (measured from the left end) chosen randomly uniformly along its length. Then you break the left part at a point Y chosen randomly uniformly along its length. In other words, X is uniformly distributed between 0 and I and, given X =x, Y is uniformly distributed between 0 and x. a. Determine E(YLX = x) and then V(YLX = x). Is E(YLX = x) a linear function of x? b. Determine fr.y) using fy(x) and frx(ylr). c. Determine fr(y).

Question
A stick is one foot long. You break it at a point X
(measured from the left end) chosen randomly
uniformly along its length. Then you break the
left part at a point Y chosen randomly uniformly
along its length. In other words, X is uniformly
distributed between 0 and I and, given X =x, Y is
uniformly distributed between 0 and x.
a. Determine E(YLX = x) and then V(YLX = x). Is
E(YLX = x) a linear function of x?
b. Determine fr.y) using fy(x) and frx(ylr).
c. Determine fr(y).
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Transcribed Image Text

A stick is one foot long. You break it at a point X (measured from the left end) chosen randomly uniformly along its length. Then you break the left part at a point Y chosen randomly uniformly along its length. In other words, X is uniformly distributed between 0 and I and, given X =x, Y is uniformly distributed between 0 and x. a. Determine E(YLX = x) and then V(YLX = x). Is E(YLX = x) a linear function of x? b. Determine fr.y) using fy(x) and frx(ylr). c. Determine fr(y).

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