tidal quartz
Suppose you have a sample of size 17 from a population with the probability density function
[ f(x \mid \theta) = \frac{c\theta}{(1+x)^{1+\theta}} ]
for ( x > 5.6 ), ( \theta > 0 ) and where ( c ) is a constant to be obtained for solving this exercise.
The method of moments estimator for ( \theta ) given the observed sample: 6.84, 6.44, 6.53, 7.29, 6.54, 6.5,
7.27, 6.36, 6.92, 8.25, 8.11, 7.5, 6.92, 7.14, 7.05, 6.85, 7.67; is equal to:
Hello, I am attempting to solve this exercise and am currently trying to determine the constant ( c ). However, I am somewhat lost at this point. We know from the properties of the probability density function (PDF) that the integral of the PDF over all ( x ) from negative to positive infinity should equal 1:
[ \int{-\infty}^{\infty} f(x) , dx = 1, ]
and this is how the constant ( c ) is typically isolated. However, I do not have a "normal" probability density function, but rather ( f(x \mid \theta) ). Therefore, I am unsure whether the integral should be with respect to ( x ) while treating ( \theta ) as a constant, or if I should use the likelihood function expression, i.e., the product from ( i = 1 ) to ( n ) of ( f(x_i \mid \theta) ):
[ \prod{i=1}^n f(x_i \mid \theta), ]
and carry out the necessary calculations for this product. I am uncertain how to find this constant ( c ); I am at a loss. :(
lone lodge
cinder agate