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8 Problems on Mathematical Statistics - Assignment 2 | MATH 304, Assignments of Mathematical Statistics

Material Type: Assignment; Class: Mathematical Statistics; Subject: Mathematics; University: Bucknell University; Term: Unknown 1989;

Typology: Assignments

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Math 304: Homework 2
Due: February 9
1. (WMS 8.8) Suppose that Y1, Y2, Y3denote a random sample from an exponential distribution
with density function
f(x) = 1
θex/θ for x > 0.
Consider the following five estimators of θ:
ˆ
θ1=Y1,ˆ
θ2=Y1+Y2
2,ˆ
θ3=Y1+ 2Y2
3,ˆ
θ4= min{Y1, Y2, Y3},ˆ
θ5=¯
Y
(a) Which of these estimators are unbiased?
(b) Among the unbiased estimators, which has the smallest variance?
2. (WMS: 9.89) It is known that the probability pof tossing heads on an unbalanced coin is either
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 yactually ovserved, what is the MLE of p?
3. Lavine, Ch2 #11
4. Lavine, Ch2 #12
5. Lavine, Ch2 #13
6. HMC 6.1.2: Let X1,...,Xnbe a random sample from a Gam(α= 3, β =θ) distribution,
0< θ < . Determine the mle of θ.
7. HMC 6.1.7: Let the table
x0 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(X3).
8. Use R to obtain a sample of size 20 from a Poi(3). Obtain a graph of the likelihood function
and the log-likelihood function. Locate the MLE on the graphs.
1

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Math 304: Homework 2

Due: February 9

  1. (WMS 8.8) Suppose that Y 1

, Y

2

, Y

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

= Y

1

θ 2

Y

1

+ Y

2

θ 3

Y

1

+ 2Y

2

θ 4

= min{Y 1

, Y

2

, Y

3

θ 5

Y

(a) Which of these estimators are unbiased?

(b) Among the unbiased estimators, which has the smallest variance?

  1. (WMS: 9.89) It is known that the probability p of tossing heads on an unbalanced coin is either

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?

  1. Lavine, Ch2 #
  2. Lavine, Ch2 #
  3. Lavine, Ch2 #
  4. HMC 6.1.2: Let X 1

,... , X

n be a random sample from a Gam(α = 3, β = θ) distribution,

0 < θ < ∞. Determine the mle of θ.

  1. HMC 6.1.7: Let the table

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).

  1. Use R to obtain a sample of size 20 from a Poi(3). Obtain a graph of the likelihood function

and the log-likelihood function. Locate the MLE on the graphs.