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)

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