4. Halloween. parade The city of New York hires you to estimate whether it will rain during the

Halloween parade. Checking past data you determine that the chance of rain is 2D%. You

model this with the random variable H with pmf Pa") = 02, Few} = '13: where R = 1 means that it rains and H = [I that it doesn't. Your rst idea is to be lazy and just

use the forecast of a certain website. Analyzing data from previous forecasts, you determine that this website is right 70% of the time. You model this with a random variable W that

satises P[W =1|s =1) = as, HI? = me = a) = are. a. 1What is the probability that the website is wrong? Unsatised with the accuracy of the website, you look at the data used for the forecast [they

are available online]. Surprisingly the relative humidity of the air is not used, so you decide to

incorporate it in your prediction in the form of a random variable H. b. Is it more reasonable to assume that H and W are independent, or that they are conditionally

independent given R? Explain why. You assume that H and W are conditionally independent given B. More research establishes

that conditioned on R = 1, H is uniformly distributed between 13.5 and 13.7, whereas conditioned

on R = I], H is uniformly distributed between [Ll and 0.6. c. Compute the conditional pmf of R given W and H. Use the distribution to determine

whether you would predict rain for any possible value of W and H. d. "What is the probability that you make a mistake?