An instructor who teaches on a whiteboard has run out of whiteboard markers and plans to buy new markers. Suppose that, for the type of marker he will order: Each marker has 70% chance of coming from a "high-quality" manufacturer and 30% chance of coming from a "low-quality" manufacturer. The lifetime (in weeks) of a marker produced by the high-quality manufacturer follows an exponential distribution with rate = .5 (i.e., Exp(.5).) Markers produced by the low-quality manufacturer are worn out at a faster rate, with lifetime following Exp(2). Therefore, the lifetime of a randomly chosen marker (i.e., the number of weeks it takes to be worn out), X, follows a discrete mixture distribution: fX (x) = .7f1(x) .3f2(x), x > 0, where f1() is the Exp(.5) density, and f2() is the Exp(2) density. (a) Generate a random sample of size 500 for X, i.e., the lifetime of 500 randomly chosen markers. Plot the histogram of empirical density, with the theoretical density, fX (), superimposed. (b) This instructor plans to buy 20 markers. He uses one marker at a time and replaces with a new one when the current one is worn out. Thus, the total time