Statistics Assignment
Statistical Inference II: J. Lee Assignment 3
Problem 1. The exponential distribution is f(x|λ) = λe−λx and E(X) = λ−1. The cumulative distribution function is F (x) = P (X ≤ x) = 1−e−λx. Three observations are made by an instrument that reports x1 = 5 and x2 = 3, but x3 is too large for the instrument to measure and it reports only that x3 > 10. (The largest value the instrument can measure is 10.0.)
(a) What is the likelihood function?
(b) What is the mle of λ?
Problem 2. Suppose that X is a discrete random variable with P (X = 1) = θ and P (X = 2) = 1 − θ. Three independent observations of X are made: x1 = 1, x2 = 2, x3 = 2.
(a) Find the method of moments estimate of θ.
(b) What is the likelihood function?
(c) What is the maximum likelihood estimate of θ?
Problem 3. George spins a coin three times and observes no heads. He then gives the coin to Hilary. She spins it until the first head occurs, and end up spinning it four times total. Let θ denote the probability the coin comes up heads.
(a) What is the likelihood function?
(b) What is the mle of θ?
Problem 4. Let X1, . . . , Xn be an i.i.d. sample from a Rayleigh distribution with parameter θ > 0 :
f(x|θ) = x θ2 e−x
2/(2θ2), x ≥ 0.
(a) Find the method of moments estimate of θ.
(b) Find the mle of θ.
Problem 5. Let X1, . . . , Xn be an i.i.d. random variables with density function
f(x|θ) = (θ + 1)xθ, 0 ≤ x ≤ 1.
(a) Find the method of moments estimate of θ.
(b) Find the mle of θ.
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