Question

A random sample Y1 , Y 2 , ⋯,Yn  is drawn from a distribution whose probability density function is given by: f (Y ) = βe− βY ,  Y  0  &  β > 0  

a). Obtain the maximum likelihood estimator (MLE) of β. (3 points)

b). Given that  ∑ n Y i = 25  , ∑n Yi2 = 50  ,  n = 50 calculate the maximum likelihood 

estimate of β.

 c). Using the same data as in part (b), test the null hypothesis that β =1against the alternative hypothesis that β ≠1at 5% level of significance

Answer 1

a

$fY(y) = c(β)y3(1 − y)β1{0<y<1},$

b

25+50+50=125

c

maximum-likelihood estimator of θ