WUSTL ESE 425 - Set 1 Solutions (11 pages)

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Set 1 Solutions



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Set 1 Solutions

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Pages:
11
School:
Washington University in St. Louis
Course:
Ese 425 - Random Processes and Kalman Filtering
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ESE 425 Spring 2018 Homework Set 1 10 problems Due Tuesday Jan 23 1 Short proofs a Prove Theorem 4 That is if both A and B are subsets of S and A B then P B P A Hint axioms 1 and 3 and set algebra b Prove Theorem 5 That is for any A S 0 P A 1 Hint axiom 2 and theorems 2 and 4 2 The prevalence of a certain disease in the general population is one case in a population of 1 000 A test for the disease gives either a positive or negative result The test has a success rate of 99 meaning of every 100 persons with the disease who are tested the test will give a positive result 99 times on average The test has a false positive rate of 2 meaning of every 100 persons without the disease who are tested the test will give a positive result 2 times on average a What is the probability that you have the disease given you are tested and the test gives a positive result b Repeat part a but for a success rate of 95 and a false positive rate of 0 1 1 in 1 000 3 The discrete random variables rvs X and Y may each take on integer values 1 3 and 5 The joint probabilities are given below Y X 1 3 5 1 3 5 1 18 1 18 1 18 1 18 1 18 1 6 1 18 1 6 1 3 a Find marginal probabilities P X and P Y for all possible values of X and Y Hint complete the table b Are X and Y independent c Find pY X Y 5 X 3 ESE 425 Spring 2018 4 The discrete rvs X and Y may each take on only the integer values 0 and 1 The total probability P X 0 0 75 The conditional probabilities of Y are given here P Y 0 X 0 0 9 P Y 1 X 0 0 1 P Y 0 X 1 0 2 P Y 1 X 1 0 8 a Fill in the numerical values of the joint and marginal probabilities of this table Y 0 1 P X X 0 pXY X 0 Y 0 pXY X 0 Y 1 pX X 0 1 pXY X 1 Y 0 pXY X 1 Y 1 pX X 1 P Y pY Y 0 pY Y 1 b Find the conditional probabilities P X 0 Y 1 and P X 0 Y 0 5 A continuous random variable X is uniformly distributed from a to b b a a Find its pdf b Find its cdf 2a b c Find the probability that X 3 d Let a 1 and b 2 Find the probabilities that X c for c and then for c 1 6 A continuous random variable x has the following pdf where is some given parameter constant such that 0 1 0 x fX x k x 1 0 otherwise Find as functions of a k b the cdf 7 A moving particle s velocity is uniformly distributed over 1 1 Let V denote the random variable that characterizes the particle s velocity Write and sketch both the cdf and pdf of V 8 The kinetic energy k of a particle is related to its velocity v by the relationship k 12 mv 2 Suppose a particle has mass m 2 and its velocity is characterized by the random variable V with cdf and pdf as found in problem 7 Let K denote the random variable characterizing the particle s kinetic energy Find and sketch the cdf and then the pdf of K ESE 425 Spring 2018 9 Consider this joint pdf x y f XY x y e 0 a b c d x 0 and y 0 otherwise Find P X Find P X Y 1 Find P X or Y 1 Find P X and Y 1 Hint Do double integrals of the joint pdf over the appropriate region in x y sketch the region of integration for each 10 Consider this joint pdf a b c d e f g h xe x y 1 f XY x y 0 x 0 and y 0 x 0 or y 0 Find fX x Find fY y Hint Laplace trick Are X and Y independent Why Find fX Y x y Find fY X y x What is the probability that x 1 given y 1 Hint Table of integrals What is the probability that y 1 given x 1 What is the probability that both x and y are 1 ESE 425 Spring 2018 Solutions 1 Short proofs a Prove Theorem 4 That is if both A and B are subsets of S and A B then P B P A Hint axioms 1 and 3 and set algebra b Prove Theorem 5 That is for any A S 0 P A 1 Hint axiom 2 and theorems 2 and 4 a Express B as B A A B where A A B 0 Then by axiom 3 P B P A P A B But by axiom 1 P A B 0 Therefore P B P A b Since 0 A S by Theorem 4 P 0 P A P S Then by Theorem 2 and axiom 2 0 P A 1 2 The prevalence of a certain disease in the general population is one case in a population of 1 000 A test for the disease gives either a positive or negative result The test has a success rate of 99 meaning of every 100 persons with the disease who are tested the test will give a positive result 99 times on average The test has a false positive rate of 2 meaning of every 100 persons without the disease who are tested the test will give a positive result 2 times on average a What is the probability that you have the disease given you are tested and the test gives a positive result b Repeat part a but for a success rate of 95 and a false positive rate of 0 1 1 in 1 000 Use Bayes Rule Let T denote the event of a positive test result and D having the disease 99 2 P T D 100 a P T D 100 999 P D 1 P D 1000 1 P D 1000 P D T b P T D P D P T D P D P T D P D 99 0 0472 4 72 999 2 100 1000 99 2 999 99 1 100 1000 99 1 100 1000 1 P T D 1000 95 P T D 100 P D T 95 1 100 1000 95 1 100 1000 999 1 1000 1000 950 0 487 48 7 950 999 ESE 425 Spring 2018 3 The discrete random variables rvs X and Y may each take on integer values 1 3 and 5 The joint probabilities are given below Y X 1 3 5 1 3 5 1 18 1 18 1 18 …


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