Math 180A - Homework 7 ( Due Thursday, Nov 10, 6:00 PM) Reading: Sections 5.1 - 5.5 of the textbook. Show FULL JUSTIFICATION for all your answers. Warm Up Questions(Do not turn in) 1. Suppose X is a continuous random variable. Prove Var(aX + b) = a2 Var(X). 2. Suppose X is a continuous random variable. Prove Var(X) = E[X 2 ] (E[X])2 . 3. Problem 5.7 of the textbook. 4. Problem 5.15 of the textbook. Homework Problems (To turn in) 1. Let F be the cumulative distribution function of a random variable X. Prove limx F (x) = 1. Hint: Follow the proof of limx F (x) = 0: show that for every sequence xn , limn F (xn ) = 1. Use the equation (1) in Problem 1 of More Practice section. 2. Suppose X is a continuous random variable with p.d.f. given by \u001a 0 if x < 1 f (x) = c if x 1. x3 (a) Find c. (b) Compute the c.d.f of X. (c) Find E[X]. (d) Find Var(X). 3. Problem 5.31 of the textbook. 4. Suppose the lifetime of a radio is a random variable T , that has a Exponential distribution with average life time of 10 years. (a) Write the probability density function of T . (b) If you buy a new radio, what is the probability that it works for at least 15 years? (c) If you buy a 10-years-old radio, what is the probability that it works for at least 15 more years? Do you prefer to buy a new radio or a 10-years-old radio? 5. Suppose the lifetime of a radio is a random variable T , with the following p.d.f. \u001a 0 if x < 5 f (x) = 50 if x 5. x3 (a) What is the average life time of the radio? 1 (b) If you buy a new radio, what is the probability that it works for at least 15 years? (c) If you buy a 10-years-old radio, what is the probability that it works for at least 15 more years? Do you prefer to buy a new radio or a 10-years-old radio? 6. Problem 5.17 of the textbook. 7. Problem 5.24 of the textbook. More Practice (Do not turn in) 1. In this problem, we want to prove the following proposition stated in class: Suppose I1 , I2 , ... are subsets of R such that I1 I2 ..... Then, lim P (In ) = P (I), n where I = [ In . (1) n=1 c . Define A1 = I1 , and for k 2, An = In \\ In1 = In In1 (a) Write In in terms of A1 , A2 , ...An . Write P (X In ) in terms of P (X A1 ),P (X A2 ),...,P (X An ). (b) Write I in terms of A1 , A2 , .... Write P (X I) in terms of P (X A1 ),P (X A2 ), .... (c) Conclude that the equation (1) holds. 2. Theoretical Exercise 5.11. 3. Theoretical Exercise 5.13. 4. Theoretical Exercise 5.14. 2