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Material Type: Assignment; Class: Mathematical Statistics; Subject: Mathematics; University: Bucknell University; Term: Unknown 1989;
Typology: Assignments
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3
denote a random sample from an exponential distribution
with density function
f (x) =
θ
e
−x/θ
for x > 0.
Consider the following five estimators of θ:
θ 1
1
θ 2
1
2
θ 3
1
2
θ 4
= min{Y 1
2
3
θ 5
(a) Which of these estimators are unbiased?
(b) Among the unbiased estimators, which has the smallest variance?
1/4 or 3/4. The coin is tossed twice and a value for Y , the number of heads, is observed. For
each possible value of Y , which of the two values for p (1/4 or 3/4) maximizes the probability
that Y = y? Depending on the value of y actually ovserved, what is the MLE of p?
n be a random sample from a Gam(α = 3, β = θ) distribution,
0 < θ < ∞. Determine the mle of θ.
x 0 1 2 3 4 5
frequency 6 10 14 13 6 1
represent a summary of a sample of size 50 from a binomial distribution having n = 5. Find
the mle of P (X ≥ 3).
and the log-likelihood function. Locate the MLE on the graphs.