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Question #1 [10 marks] Benford's law states that in many naturally occurring numbers that the leading significant digit tends to be 'sTall' (e.g. stock prices, city populations). In these scenarios, the leading digit is a discrete random variable X with probability mass function f(x) = logk (1 + E) where k is dependent on the number system in use (e.g. binary, decimal, etc.). For remainder of this question, suppose that a; E {1, 2, 3} (Le. the quaternary number system is being used which can be used to represent genetic code like DNA). a. Using properties of logarithms, determine the exact value of k. b. State the p.m.f. of X as a table. Results may be rounded to 4 decimal places (where necessary). c. State the c.d.f. of X as a table. Results may be rounded to 4 decimal places (where necessary). [Note: Use interval notation for (3.] F(x) - d. Showing your work, determine px = E[X], the expected value (or mean) of X. Result may be rounded to 4 decimal places. e. Showing your work, determine a}, = E[XZ] #2, the variance in X. Result may be rounded to 4 decimal places