7/25/2019 Problem+Set+5+Solutions

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(1525)1

2 t5

V(T1) =V(X)/n2 =npq/n2 =pq/n

V(T2) =V(X+ 1)/(n+ 2)2 =V(X)/(n+ 2)2 =npq/(n+ 2)2

E(T1) =E(X)/n= np/n= p (E(T1)p)2

= 0E(T2) =E(X+ 1)/(n+ 2) = (E(x) + 1)/(n+ 2) = (np+ 1)/(n+ 2)(E(T2)p)

2 = ((12p)/(n+ 2))2

MSE(T1) =pq/n

MSE(T2) =npq/(n+ 2)2 + ((12p)/(n+ 2))2 = =

1 + (np)pq

(n+ 2)2

MSE(T1) =.0024 MSE(T2) =.0023936947 MSE(T2)

E(T1) = (1/6)6

1E(X) = (1/6)(6) =

E(T2) = (1/5)6

2E(X) = (1/5)(5) =

V ar(T1) =2/6 V ar(T2) =

2/5

MSE(T3) =2V ar(T1) + (1)

2V ar(T2) =2(2

6) + (1)2

2

5MSE(T3) = 2(

2

6) 2(1)

2

5

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6

=1

5

= 6

11

MSE(T3)

L(= 1/2 |HTT) = (1/2)(1/2)(1/2) = 1/8L(= 1/3 |HTT) = (2/3)(1/3)(1/3) = 2/27L(= 2/3 |HTT) = (1/3)(2/3)(2/3) = 4/27 2/3

ln(L) =7 + (xi)ln()20 + (yi)ln() + ( xi! yi!)

=xi+ yi

27

V ar() =

V ar(xi+ yi

27 ) = (V ar(xi) + V ar(yi))/27

2 = (7+ 10 2)/272 = /27

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