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decide Tthe problern is easy? Problem 2 (The Weather frog) Let us assume that a weather frog bases his forecast for tomorrows weather entirely on todays air pressure. Determining a weather forecast is a hypothesis testing problem. For simplicity, let us assume that the weather frog only needs to tell us if the forecast for tomorrows weather is sunshine or rain. Hence we are dealing with binary hypothesis testing. Let H 0 mean sunshine and H 1 mean rain. We will assume that both values of H are equally likely, i.e. PH(0) = PH (1 ) : . Measurements over several years have led the weather frog to conclude that on a day that pre- cedes sunshine the pressure may be modeled as a random variable Y with the following probability density function: 0, otherwise. YlH(у to search

Similarly, the pressure on a day that precedes a rainy day is distributed according to 0, otherwise The weather frogs goal in life is to guess the value of H after measuring Y (a) Determine A and B. (b) Find the a posteriori probubility Py (Oly). Also find Paly (1l). Hint: Use Bayes rule. (e) Plot PEY (Oly) and PHİY(1 y) as a function of y Show that the implementation of the decision rule H(u) arg max, PHy (l) reduces to 11e(y) İl: otherwise, for some threshold θ and specify the thresholds value. (d) Now assume that you implement the decision rule (u) and determine, as a function of y, (e) For the same decision rule, determine the propability of error P.() as a function of . f) Using calculus, find the that minimizes P (7) and compare your result to θ Could you the probability that the decision rule decides H 1 given that H 0. This probability is denoted Pr) 1H 0 Evaluate your expression at y 6 lype here to search

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