5. (15 points) Let X be a random variable with cumulative distribution function 0 x1 a. Find P(-1 2X, . b. Find the marginal probability density function of X, . c. Find EX,X2. d. Find EXI. 9. (10 points) The random variable X has moment generating function M(t) =et(e-1). What are the mean and variance of this random variable? 10. (10 points) An executive must decide between two competing techniques to complete a given task. Technique A will work only 15% of the time, but only costs $300. Technique B will work 55% of the time and costs $5000. The executive must balance time versus cost, so he decides on the following rule: Try technique A for a set number of times, k. If it doesn't work, switch to technique B and keep trying that until success occurs. He must therefore find the expected number of attempts and the expected cost for various values of the variable k. a. Give a probability mass function for the number of attempts if k=3. b. Write, but do not evaluate, the formula for expected number of attempts with k=3. c. Give a probability mass function for the cost of success if k=2. d. Write, but do not evaluate, the formula for the expected cost of success for k=2. e. (5 points extra credit) Evaluate the formulas you found in b and d