120B-HW1-W19

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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 ).
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