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Stats 120B Homework 1: Due Thursday, Jan. 24 1. Let X and Y be two independent random variables following beta distributions Beta(120, 2019). (a) What’s P(X = 0.3)? (b) What’s E(2X − Y )? (c) What’s P(2X + 4 > 3Y )? (d) What’s P(X < Y )? (e) Now if X and Y are no longer independent to each other. Will the answers to (a)–(d) remain the same? Explain. (f) Now define Z ∼ Beta(2019, 120). Compare the median of X and Z, which one is bigger? Compare the variance of X and Z, which one is bigger? Explain. (Hint: no calculation is needed) 2. Let I(·) be the indicator function. Prove IA∩B = IA × IB = min{IA , IB } for any two sets A and B. 3. Suppose that X has the gamma distribution with parameters α and β. (a) Determine the mode of X. (Be careful about the range of α) (b) Let c be a positive constant. Show that cX has the gamma distribution with parameters α and β/c. 4. Suppose that Xi ∼ Gamma(αi , β) independently for i = 1, . . . , N . The mgf of Xi is MXi (t) = (1 − βt )−αi . P (a) Use the mgf of Xi to derive the mgf of N i=1 Xi . PN (b) Determine the distribution of i=1 Xi based on its mgf. PN iid (c) Suppose that Y1 , . . . , YN ∼ Exp(β), determine the distribution of i=1 Yi using results from part (b). P (d) Determine the distribution of sample mean Ȳ = N i=1 Yi /N , using results from problem 3. 5. Let X follow beta distribution Beta(α, β). (a) Determine the mode of X. (Be careful about the range of α and β) (b) Show that 1 − X has the beta distribution with parameters β and α. 6. Suppose that a random variable X can take each of the five values −2, −1, 0, 1, 2 with equal probability. Determine the probability mass function of Y = |X| − X. 7. Suppose that X1 , . . . , Xk are independent random variables, and PkXi ∼ Exp(βi ) for i = 1, . . . , k. Let Y = min{X1 , . . . , Xk }. Show that Y ∼ Exp( i=1 βi ). 1