Exercise 3.61. Let the value of your car be M dollars. (M can be, for example,...

 

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Exercise 3.61. Let the value of your car be M dollars. (M can be, for example, 2000 but it is easier to write mathematics with M than with long numbers.) When you get into an accident, the amount of damage to your car is X dollars, where X is a random variable with probability density function f(x) = 2(M-x)/M, 0 x M 0, x <0 or r > M. You have insurance with an M/2 dollar deductible. This means that if X M/2 then you pay M/2 dollars and the insurance company covers the remaining X - M/2 dollars. Let Y be the amount you pay for the damage. (a) Find the cumulative distribution function of X. (b) Give a formula that expresses Y as a function of X. (c) Find the probabilities P(Y y) for y < M/2, P(Y = M/2), and P(Y y) for yM/2. Based on these, write a formula for and sketch the graph of the cumulative distribution function Fy of the random variable Y. (d) Find the probability P(Y < M/2). How can this probability be found from the c.d.f. Fy? (e) Does Y qualify either as a discrete or as a continuous random variable? Hint. When you write down probabilities, be careful with matters. If you need inspiration, look at Example 3.20. and < when it Exercise 3.61. Let the value of your car be M dollars. (M can be, for example, 2000 but it is easier to write mathematics with M than with long numbers.) When you get into an accident, the amount of damage to your car is X dollars, where X is a random variable with probability density function f(x) = 2(M-x)/M, 0 x M 0, x <0 or r > M. You have insurance with an M/2 dollar deductible. This means that if X M/2 then you pay M/2 dollars and the insurance company covers the remaining X - M/2 dollars. Let Y be the amount you pay for the damage. (a) Find the cumulative distribution function of X. (b) Give a formula that expresses Y as a function of X. (c) Find the probabilities P(Y y) for y < M/2, P(Y = M/2), and P(Y y) for yM/2. Based on these, write a formula for and sketch the graph of the cumulative distribution function Fy of the random variable Y. (d) Find the probability P(Y < M/2). How can this probability be found from the c.d.f. Fy? (e) Does Y qualify either as a discrete or as a continuous random variable? Hint. When you write down probabilities, be careful with matters. If you need inspiration, look at Example 3.20. and < when it