Let $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ represent a random sample from a Rayleigh distribution with pdf $f ( x ; \theta ) = \frac { x } { \theta } e ^ { - x ^ { 2 }/ ( 2 \theta ) }\quad x 0$
a. It can be shown that $E \left( X ^ { 2 } \right) = 2 \theta$
Use this fact to construct an unbiased estimator of $\theta$
based on $\sum x _ { i } ^ { 2 }$
(and use rules of expected value to show that it is unbiased).
b. Estimate $\theta$
from the following $n = 10$
observations on vibratory stress of a turbine blade under specified conditions:
17.08 10.43 4.79 6.86 13.88
14.43 20.07 9.60 6.71 11.15

Question

Let $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ represent a random sample from a Rayleigh distribution with pdf $f ( x ; \theta ) = \frac { x } { \theta } e ^ { - x ^ { 2 }/ ( 2 \theta ) }\quad x > 0$ 
a. It can be shown that $E \left( X ^ { 2 } \right) = 2 \theta$ 
Use this fact to construct an unbiased estimator of $\theta$ 
based on $\sum x _ { i } ^ { 2 }$ 
(and use rules of expected value to show that it is unbiased). 
b. Estimate $\theta$ 
from the following $n = 10$ 
observations on vibratory stress of a turbine blade under specified conditions:
17.08 10.43 4.79 6.86 13.88
14.43 20.07 9.60 6.71 11.15

Quizplus · Accepted Answer

The Answer of Let $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ represent a random sample from a Rayleigh distribution with pdf $f ( x ; \theta ) = \frac { x } { \theta } e ^ { - x ^ { 2 }/ ( 2 \theta ) }\quad x > 0$ 
a. It can be shown that $E \left( X ^ { 2 } \right) = 2 \theta$ 
Use this fact to construct an unbiased estimator of $\theta$ 
based on $\sum x _ { i } ^ { 2 }$ 
(and use rules of expected value to show that it is unbiased). 
b. Estimate $\theta$ 
from the following $n = 10$ 
observations on vibratory stress of a turbine blade under specified conditions:
17.08 10.43 4.79 6.86 13.88
14.43 20.07 9.60 6.71 11.15

Let X1,X2,…,XnX _ { 1 } , X _ { 2 } , \ldots , X _ { n }X1​,X2​,…,Xn​

Let $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$