9. Suppose that a point is chosen at random on a stick of unit length and that the stick is broken into two pieces at that point. Find the expected value of the length of the longer piece.

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8. Suppose that X and Y have a continuous joint distribu-
tion for which the joint p.d.f. is as follows:
f(x, y) = { 12y- for 0 < y < x < 1,
otherwise.
Find the value of E(XY).
9. Suppose that a point is chosen at random on a stick of
unit length and that the stick is broken into two pieces at
that point. Find the expected value of the length of the
longer piece.
10. Suppose that a particle is released at the origin of
the xy-plane and travels into the half-plane where x > 0.
Suppose that the particle travels in a straight line and that
the angle between the positive half of the x-axis and this
line is a, which can be either positive or negative. Suppose,
finally, that the angle a has the uniform distribution on the
interval [-x/2, x/2]. Let Y be the ordinate of the point at
which the particle hits the vertical line x = 1. Show that
the distribution of Y is a Cauchy distribution.
11. Suppose that the random variables X1, .
a random sample of size n from the uniform distribution
on the interval [0, 1]. Let Y1
Y, = max{X1, ..., X,}. Find E(Y1) and E (Y„).
form
min{X1, .
X,}, and let
12. Suppose that the random variables X1,..., X, form
a random sample of size n from a continuous distribution
for which the c.d.f. is F, and let the random variables Y,
and Y, be defined as in Exercise 11. Find E[F(Yj)] and
E[F(Y„)].
13. A stock currently sells for $110 per share. Let the price
of the stock at the end of a one-year period be X, which will
take one of the values $100 or $300. Suppose that you have
the option to buy shares of this stock at $150 per share
at the end of that one-year period. Suppose that money
4.2 Properties of Expectations
217
231
c. Consider the same transactions as in part (a), but
this time suppose that the option price is $x where
x > 20.19. Prove that our investor gains 4.16x – 84
dollars of net worth no matter what happens to the
stock price.
911
The situations in parts (b) and (c) are called arbi-
Transcribed Image Text:8. Suppose that X and Y have a continuous joint distribu- tion for which the joint p.d.f. is as follows: f(x, y) = { 12y- for 0 < y < x < 1, otherwise. Find the value of E(XY). 9. Suppose that a point is chosen at random on a stick of unit length and that the stick is broken into two pieces at that point. Find the expected value of the length of the longer piece. 10. Suppose that a particle is released at the origin of the xy-plane and travels into the half-plane where x > 0. Suppose that the particle travels in a straight line and that the angle between the positive half of the x-axis and this line is a, which can be either positive or negative. Suppose, finally, that the angle a has the uniform distribution on the interval [-x/2, x/2]. Let Y be the ordinate of the point at which the particle hits the vertical line x = 1. Show that the distribution of Y is a Cauchy distribution. 11. Suppose that the random variables X1, . a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 Y, = max{X1, ..., X,}. Find E(Y1) and E (Y„). form min{X1, . X,}, and let 12. Suppose that the random variables X1,..., X, form a random sample of size n from a continuous distribution for which the c.d.f. is F, and let the random variables Y, and Y, be defined as in Exercise 11. Find E[F(Yj)] and E[F(Y„)]. 13. A stock currently sells for $110 per share. Let the price of the stock at the end of a one-year period be X, which will take one of the values $100 or $300. Suppose that you have the option to buy shares of this stock at $150 per share at the end of that one-year period. Suppose that money 4.2 Properties of Expectations 217 231 c. Consider the same transactions as in part (a), but this time suppose that the option price is $x where x > 20.19. Prove that our investor gains 4.16x – 84 dollars of net worth no matter what happens to the stock price. 911 The situations in parts (b) and (c) are called arbi-
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