Tutorialweek8

.pdf
School
University of New South Wales **We aren't endorsed by this school
Course
MATH 2070
Subject
Mathematics
Date
Oct 28, 2023
Pages
1
Uploaded by CorporalIceHerring35 on coursehero.com
UNIVERSITY OF SYDNEY School of Mathematics and Statistics MATH2070/2970 Optimisation and Financial Mathematics Advanced questions are marked A . Week 8 Question 1. Let X be the sum of the numbers showing when two dice are thrown. Give the probability distribution of X and calculate its mean and variance. Question 2 ( A ) . The binomial distribution is a discrete distribution defined by the probability function b ( k ; n, p ) = n k p k q n k ; 0 k n and represents the probability of k successes in n independent trials when p is the probability of a success in a single trial and q = 1 p . Using direct computations show that E [ X ] = np and Var ( X ) = npq , where Var ( X ) denotes the variance of X . Question 3. The yield Y of a zero-coupon bond is assumed to be a continuous random variable, expo- nentially distributed with p.d.f. (probability density function) given by f ( y ) = 1 µ e y/µ ; y 0 . Find the expected value and variance of the yield. Question 4. Given that the continuous compound rate R is normally distributed with mean µ and variance σ 2 , show that the expected discount factor is E [ e Rt ] = e ( µ σ 2 t/ 2) t . Question 5. Let X be a random variable uniformly distributed on the interval [ a, b ] (with a < b ). Compute the cumulative distribution function, expectation and variance of X . Recall that the p.d.f. of the uniform distribution is f ( x ) = ( 1 b a if x [ a, b ] 0 if x / [ a, b ] . Question 6. Let X be a random variable. Find the minimum of γ ( t ) = E [( X t ) 2 ] . What is the minimiser? Question 7. (Linear regression) Let X and Y be two random variables. We aim to approximate Y by a linear function of X . Find a and b which minimise the square error function e ( a, b ) = E [( Y aX b ) 2 ] .
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