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(Solved): The pdf of the log-normal distribution LN(\mu ,\sigma ^(2)) is given by: f(x|\mu ,\sigma ^(2))=(1)/ ...
The pdf of the log-normal distribution
LN(\mu ,\sigma ^(2))is given by:
f(x|\mu ,\sigma ^(2))=(1)/(x\sigma \sqrt(2\pi ))exp{-((logx-\mu )^(2))/(2\sigma ^(2))},x>0,\mu inR,\sigma >0;given a random sample
{x_(i)}_(i)=1^(n)?^(i.i.d.)LN(\mu ,\sigma ^(2))(a) Compute the maximal likelihood estimators for the mean and the variance
(hat(\mu )_(MLE),hat(\sigma )_(MLE)^(2));
(4pt)(b) Derive the Fisher Information
I(\mu )and
I(\sigma ^(2));(5pt)(c) Find pivots and construct asymptotic confidence intervals for
\mu and
\sigma ^(2)(8 pt);
***A random sample of
n=20students gave ACT scores
x: Some calculations include that
\sum_i x_(i)=501, and
\sum_i x_(i)^(2)=12687.50, then derive a two-tailed
95%confidence interval for both
\sigma ^(2)and
\sigma (8pt)(Hint: if you derive a CI for one of them, then the other's CI is immediate)
