20.9 Problems

Problem 1. Let X,Y : Ω →ℝ be two random variables.

(a) Show that if X ≥ 0, then 𝔼[X] ≥ 0.

(b) Show that if X ≥Y , then 𝔼[X] ≥𝔼[Y ].

Problem 2. Let X : Ω →ℝ be a random variable. Show that if Var[X] = 0, then X assumes only a single value. (That is, the set X(Ω) = {X(ω) : ω ∈ Ω} has only a single element.)

Problem 3. Let X ∼ Geo(p) be a geometrically distributed (Section 19.2.3) discrete random variable. Show that

Hint: Use that for any q ∈ (0,1), ∑ _k=1^∞kq^k−1 = (1 −q)⁻².

Problem 4. Let X ∼ exp(λ) be an exponentially distributed continuous random variable. Show that

Problem 5. Find the maximum likelihood estimation for the λ parameter of the exponential distribution.

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