## He horn of an auto operates on demand 99% of the time. Assume each time you hit the horn, it works or fails in?

### He horn of an auto operates on demand 99% of the time. Assume each time you hit the horn, it works or fails in?

The horn of an auto operates on demand 99% of the time. Assume each time you hit the horn, it works or fails independently of all other times.(a) How many times do you expect to be able to honk the horn with 75% probability of not having any failures.(b) What is the expected number of times you hit the horn before the tenth failure?

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Show that if T has the Weibull (¦E, ¦A) distribution with the following density: f (t) = ¦E¦At¦A1 e¦Et¦A(t > 0),where ¦E > 0 and ¦A > 0 then T ¦A has an exponential ¦E distribution. [b (10 points)]. Show that if U is uniform (0, 1) random variable then T = (¦E1 log U )1/¦A has a Weibull (¦E, ¦A) distribution. 3. Let Y be the minimum of 4 independent random variableswith uniform distribution on (0, 1) and let Z be their maximum. Find: [a (10 points)]. P (Z?U 3/4|Y ?Y 1/4). [b (10 points)]. P (Z ?U 3/4|Y?U 1/4). 4. Insurance claims arrive at an insurance company according to a Poisson process with rate ¦E. The amount of each claim has an exponential distribution with rate independently of times and amounts of all other claims. Let Xt denote the accumulated total of claims between time 0 and time t. Find simple formulae for [a (3 points)]. E(Xt ). 2 [b (5 points)]. E(Xt ). [c (5 points)]. SD(Xt ). [d (7 points)]. Corr(Xt , XS ) fort < s. 5. Let X1 , X2 , X3 , X4 be independent random variables with distribution Exp(¦E). Let Xmin denote the minimum of the X??s andXmax denote the maximum. [a (10 points)]. For a a, Xmax < b). [b (10 points)]. What is the joint density of Xmin andXmax ? 6. Let X and Y be the scores of the midterm and nal exams. Suppose that E[X] = 60 and E[Y ] = 70 and that all scores are between 0 and 100. [a (8 points)]. How can you upper bound P [X ?U 40, Y ?Y 90]? Is your bound optimal? [b (8 points)]. Suppose that in addition you are also given that (X, Y ) is bivariate normal with ¦N = 0.9 and SD(X) = SD(Y ) = 10. Find E[X|Y ?Y 90] and bound usingMarkov inequality P [X ?U 40|Y ?Y 90]. [c (6 points)]. Use the previous bound to obtain anupper bound on P [X ?U 40, Y ?Y 90] when (X, Y ) is bivariate normal with ¦N = 0.9 and SD(X)= SD(Y ) = 10.

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