MAT 4872 Midterm Review Spring 2007 Prof. S. Singh

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1 MAT 487 Midterm Review Spring 007 Prof S Singh Answer all questions carefully and give references for material used from journals Sketch all state spaces for all transformations You are required to work indepently 1 Let X and Y be continuous random variables with densities f X ( x ) and f ( y ), respectively, and suppose that X and Y are Y indepent Use the transformation U = X + Y, V = X, to show that the density of U satisfies the convolution equation: fu ( u) = fx ( x) fy ( u v) dv If X and Y are iid uniform distributions on, find the distribution of U = X + Y (0,1) ( ) mod1 We will examine the quadratic equation Ax + Bx+ C, where A, B and C are indepent random variables, uniformly distributed on (0,1) a Find the distribution of X = lnb b Find the distribution of Y = ln A ln C c Explain why X and Y are indepent d Show that the probability that the quadratic equation have real roots is equivalent to solving PY ( X ln 4) e Evaluate: PY ( X ln 4) f Write a program to simulate the values of A, B and C Your program must count the number of times that your simulated triple, ( A, BC, ) generate real roots to the quadratic equation Run your program for ten thousand trials and compare the probability obtained with your answer in part e

2 X Let X N 3, Define Y1 = 5X1 Xand X Y1 Y = X + 3X3 Find the moment generating function of Y 4 The random variables and Y are iid, both with density f( y) = y y ( for 1, ) Y1 and zero elsewhere Find the joint density of and U, where U1 density of U 1 U 1 Y1 = Y + Y 1 and U = Y1 + Y Find the marginal 5 A bivariate distribution is always normal if its marginals are normal Prove or give a counterexample to this claim Justify all steps in this solution

3 Problems and solutions 1 Let X and Y be continuous random variables with densities f X ( x ) and f ( y ), respectively, and suppose that X and Y are Y indepent Use the transformation U = X + Y, V = X, to show that the density of U satisfies the convolution equation: fu ( u) = fx ( x ) fy ( u v) dv If X and Y are iid uniform distributions on ( 0,1 ), find the distribution of U = ( X + Y) mod1 The first part is a direct computation U is a uniform distribution on ( 0,1 ) Explain this carefully We will examine the quadratic equation Ax + Bx+ C, where A, B and C are indepent random variables, uniformly distributed on (0,1) g Find the distribution of X = lnb 1 x / X e for x ( 0, ) and zero elsewhere This is an exponential distribution h Find the distribution of Y = ln A ln C y Y ye for and zero elsewhere This is a gamma distribution y (0, ) i Explain why X and Y are indepent This is clear Explain j Show that the probability that the quadratic equation PY X ln 4 have real roots is equivalent to solving ( ) Start with B 4AC and take logs of both sides

4 k Evaluate: PY ( X ln 4) { xy, 0 x,0 y } The region ( ) region {( uv, ) 0 v, v u} U = Y X and V = Y u / e for u,0 9 f( u) = 1 u u+ e for u 0, 3 3 < < gets mapped into the < < by the transformation: ( ) ( ) ( X ln 4) = ( ln 4) PY 1 u 5 1 PU = u e du ln ln 4 + = l Write a program to simulate the values of A, B and C Your program must count the number of times that your simulated triple, ( A, BC, ) generate real roots to the quadratic equation Run your program for ten thousand trials and compare the probability obtained with your answer in part e The matlab program below simulates the triple ( A, BC, ) ; finds the probability of real roots; and gives a graphical illustration of the distribution of the real roots in red and the imaginary roots in green function qroot(n) counter=0; for i=1:n A=rand; B=rand; C=rand; D=B^-4*A*C; if (D>=0) counter=counter+1; plot(i,d,'r') hold on else plot(i,d,'g') hold on

5 probability = counter/n Probability= 0579 for 10,000 triples, which is close to the theoretical value Here is an alternative way to solve the above problem Solution II For imaginary roots: c > b /4a(See the shaded region above), if b is fixed, then b /4a < c< 1, b /4< a < 1 from the above diagram Clearly 0< b < 1, therefore the probability of real roots is:

6 b /4 b /4a 5 1 dc da db = + ln 36 6 The above solution is much more nicer and it can be exted to consider all quadratics If we consider ab, and c to be uniform on ( 1,1), then imaginary roots can be found when a and c have the same sign In the above analysis we consider the case a > 0 and c > 0 The joint density function of ab, and c is 1/ 8 on the cube of side units centered around the origin Now the probability of real roots is: dc da db = + ln b /4 b /4a A simulation run by matlab with the program qroot1 gives: qroot1( ) probability = function qroot1(n) counter=0; for i=1:n A=*rand-1; B=*rand-1; C=*rand-1; D=B^-4*A*C; if (D>=0) counter=counter+1; probability = counter/n X Let X N 3, Define Y1 = 5X1 Xand X Y1 Y = X + 3X3 Find the moment generating function of Y / 1 / M () t = exp t μ + t t) = exp(45t1 + 84t1t + 49t + 4t t 1 y1 y

7 4 The random variables and Y are iid, both with density f( y) = y ( for 1, y ) Y1 and zero elsewhere Find the joint density of and U, where U1 density of U 1 U 1 Y1 = Y + Y 1 and U = Y1 + Y Find the marginal f u ( u) 1 1 for u 0, 1 ( ) u = 1 1 for u,1 u 5 A bivariate distribution is always normal if its marginals are normal Prove or give a counterexample to this claim Justify all steps in this solution ( x + y ) xye 05( x + y ) Consider the joint density f( x, y) = e, π where x and y, for a counterexample

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