AMA563
Assignment1. Due: Oct 12, 2020 (Monday), 6:30pm.
1. Suppose X follows an exponential distribution with pdf fX(x) = λe
−λx on x > 0, and
zero elsewhere. Find the moment generating function of X. Find the expectation of E 1√
X
.
2. Let X1, . . . , Xn be a random sample from the pdf f(x|θ) = 1θ−1 on 1 ≤ x ≤ θ and zero
elsewhere, with θ > 1.
a. Find the method of moments estimator θ̂MME for θ.
b. Find the maximum likelihood estimator θ̂2MLE for θ
2.
c. Find the bias of θ̂2MLE.
3. Let X1, . . . , Xn be a random sample from the pdf f(x|θ) = θ(1−x)θ−1, 0 < x < 1, θ > 0,
zero elsewhere.
a. Find the method of moments estimator θ̂MME for θ.
b. Find the maximum likelihood estimator θ̂MLE for θ.
c. Compute the Fisher information I(θ).
4. Let X1, . . . , Xn be a random sample from the pdf f(x|θ) = −(θ + θ2)(1 + x)θ−1x, −1 <
x < 0, θ > 0, zero elsewhere.
a. Find the method of moments estimator θ̂MME for θ.
b. Find the maximum likelihood estimator θ̂MLE for θ.
c. Compute the minimum possible variance of all the unbiased estimators for θ2.
5. Let X1, . . . , Xn be a random sample from Poisson(λ).
a. Find the maximum likelihood estimator λ̂MLE for λ.
b. Compute the Fisher information I(λ).
c. Compute the bias of (X)2 as an estimator for λ2.
d. Compute the minimum possible variance of all the unbiased estimators for 1λ .
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