A01.1 Discrete pmfs to edfs Obtain the cdf of the following discrete random vari- ables (i) R where for some parameter, 0 c (0, 1), P( R = 0) = 02, P(R = 1) = 0(1 - 0), P(R = 2) = 0(1 - 0). P(R = 3) = (1 - 0), and for re {0. 1, 2, 3), P(R = r) = 0. (ii) R - Unif(0, m). [4] Hint: The discrete uniform pmf is given near the start of Section 4 of your Math230 notes and in your first year notes. Study Figure 2.1 in your notes. Your function should work for any real number 7 6 R. First calculate F at the integers, being careful to distinguish the index of summation from the upper limit of summation. Secondly extend from the integers to the real line, either by stating ranges or using the int function, where relevant. A01.2 Butterfly The lifetime, X, in days of a species of butterfly has a density of fx ( x) =1 b /2 131 . (i) Find the cdf of X in terms of b, being careful to show your working. (ii) Write down the value of b and explain how you deduced this. (iii) Find the probability that (i) a butterfly lives for more than two days, (ii) a butterfly lives for more than half a day. (iv) Find the age which only 1% of butterflies reach. [6] A01.3 Challenge A random variable X has the following cdf. Fx(x) = 121. By considering lim, Fx(x) - Fy(x - 1/i), or otherwise, describe the random variable itself. [3]