7/24/2019 MATH_3075_3975_t2

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MATH3075/3975

Financial MathematicsSchool of Mathematical Sciences

University of SydneySemester 2, 2012

Tutorial sheet: Week 2

Exercise 1  Assume that joint probability distribution of the two-dimensionalrandom variable (X, Y  ), that is, the set of probabilities

P(X  = i, Y   = j) = pi,j   for  i, j  = 1, 2, 3,

is given by: p1,1 = 1/9, p2,1  = 1/3, p3,1 = 1/9, p1,2 = 1/9, p2,2  = 0, p3,2  = 1/18, p1,3 = 0, p2,3  = 1/6, p3,3 = 1/9.

(a) Compute  EP(X | Y  ), that is,  EP(X |Y   = j) for  j  = 1, 2, 3.(b) Check that the equality  EP(X ) =  EP[EP(X | Y  )] holds.(c) Are the random variables  X   and  Y   independent?

Exercise 2  The joint probability density function f (X,Y  ) of random variablesX  and  Y   is given by

f (X,Y  )(x, y) = 1

y e−

x/ye−

y,   ∀ (x, y) ∈  R2+,

and  f (X,Y  )(x, y) = 0 otherwise.(a) Check that  f (X,Y  )  is a two-dimensional probability density function.(b) Show that  EP(X | Y   = y) = y   for all  y ∈  R+.

Exercise 3   Let  X   be a random variable uniformly distributed over (0, 1).Compute the conditional expectation  EP(X |X < 1/2).

Exercise 4   Let   X   be an exponentially distributed random variable withparameter λ > 0, that is, with the probability density function  f X (x) =   1

λe−

x

λ

for all  x > 0. Compute the conditional expectation  EP(X |X > 1).

Exercise 5  We assume that

P(X  = ±1) = 1/4,   P(X  = ±2) = 1/4

and we set   Y   =   X 2. Check whether the random variables   X   and   Y   arecorrelated and/or dependent.

Exercise 6   (MATH3975) Let  U   and  V  have the same probability distribu-tion and let   X   =   U   + V    and   Y   =   U  − V   . Examine the correlation andindependence of the random variables  X  and Y  (provide relevant examples).

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