Question

Problem 3

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.

b). Given that Σ n

Y i = 25 ,

Σn

Yi2

= 50 , n = 50 calculate the maximum likelihood

i =1

i =1

estimate of β.

(3 point)

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) maximum likelihood estimator.

$\frac{df}{dy} = \beta$$(\beta e$$-\beta$$Y)-$$\beta$($\beta$$e-\beta$$Y)$

(b)

M= $\frac{1}{n}\sum$ X_i (from 1to 50)

n= 50

X_i= 1

M=$\frac{1}{50}= 0.02$

(c)T²$=\frac{1}{n}\sum$(Xi -M)²

$=0.02(1-0.02)= 0.0196$