# WUSTL ESE 425 - Set 1 Solutions (11 pages)

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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

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