1. (10 points) The probability density function of a random variable X is given by The pdf is, f(:I:;k) = O-xk :1:), ifO S :1: S 1 where k > 0 is known. and f (:13; k) = 0, otherwise. a. What is the value of constant C (in terms of k)? b. Compute P(X S b) for b > 1. 2. (8 + 4 points) An ecologist wishes to mark off a rectangular sampling region having each side 5 meters long. However, actual length of each side, say L is a random variable due to measurement error. The length L is thus given by L = g + X where X has the pdf given in Problem 1 above with value of k = 2.(Write down the pdf explicitly.) Clearly when a square is desired then horizontal and vertical sides both must be L and thus the area will then be L2. i. What is the expected area of the resulting square region? ii. Now suppose the lengths of horizontal side and vertical side, say L1 and L2 respectively are independently chosen from the above pdf. Thus L1 and L2 are tin! and the area of the rectangular region is L1L2. What is the expected area of this resulting rectangular region