7Ct15 question

1. Consider two random variables X and Y that take values in set {1,2, ... ,n). a) Give the definition of the conditional probability P(Y=y|X=x). [5 points] b) Prove the marginalization property that P(X = x) = Et, P(X = x and Y = 1) [10 points] c) Using the marginalization property in (b) prove that )_, P(Y = i[X = x) = 1 [10 points] d) A bag of insects contains 10 crickets and 5 spiders. I draw two insects from the bag, without replacement. Given that the second insect was a spider, what is the probability that the first insect drawn was also a spider ? Hint: use marginalisation to calculate the probability that the second insect was a spider. [5 points] 2. a) State Bayes Rule. [5 points] b) Suppose 48% of people in the population support presidential candidate T and the rest support candidate C. When asked which candidate they support, 75% of supporters of candidate T answer truthfully, the others falsely answering that they support C. When asked, 100% of supporters of C answer truthfully. Suppose a person answers that they support candidate C. What is the probability that in fact they support candidate T. [10 points] 3. Five people play the game of "odd-man-out" to determine who pays for a meal. In this game, each person flips a coin. If one person's coin comes up different to all others (i.e. there is one H and four T's or there is one T and four H's), then that person pays. Otherwise, everyone flips again. They go on doing this until someone is chosen. a) What is the probability in each round that one person's coin comes up different to all others ? [5 points] b) Define the expectation of a random variable. [5 points] c) What is the expected number of times they must flip before they know who should pay? Hint pixi =- (1-x)? [10 points] b) Write a short matlab simulation of this game that outputs the number of flips made. [10 points]1. Consider two random variables X and Y that take values in set {1,2, ... ,n). a) Give the definition of the conditional probability P(Y=y|X=x). [5 points] b) Prove the marginalization property that P(X = x) = >t_, P(X = x and Y = i) [10 points] c) Using the marginalization property in (b) prove that Et_, P(Y = i|X = x) = 1 [10 points] d) A bag of insects contains 10 crickets and 5 spiders. I draw two insects from the bag, without replacement. Given that the second insect was a spider, what is the probability that the first insect drawn was also a spider ? Hint: use marginalisation to calculate the probability that the second insect was a spider. [5 points] 2. a) State Bayes Rule. [5 points] b) Suppose 48% of people in the population support presidential candidate T and the rest support candidate C. When asked which candidate they support, 75% of supporters of candidate T answer truthfully, the others falsely answering that they support C. When asked, 100% of supporters of C answer truthfully. Suppose a person answers that they support candidate C. What is the probability that in fact they support candidate T. [10 points] 3. Five people play the game of "odd-man-out" to determine who pays for a meal. In this game, each person flips a coin. If one person's coin comes up different to all others (i.e. there is one H and four T's or there is one T and four H's), then that person pays. Otherwise, everyone flips again. They go on doing this until someone is chosen. a) What is the probability in each round that one person's coin comes up different to all others ? [5 points] b) Define the expectation of a random variable. [5 points] c) What is the expected number of times they must flip before they know who should pay? Hint Epix -1 =- (1-x)2 [10 points] b) Write a short matlab simulation of this game that outputs the number of flips made. [10 points]