# Thread: Simplifying product of to sum of in likelihood equation

1. ## Simplifying product of to sum of in likelihood equation

Hi, I'm having problems with the veru bsics so I cant get any further on my questions. When given a function f(xlt). to find the likelihood you have to do the product of it. I dont understand how the product of sign changes to the summation sign.

For example L(t)= product of( t^x)xe^(-tx)

How do i simplify this to the sum of form?

Im sorry about how its written I dont know how to put in the symbols.

2. 1) It's a product of $f(x_i)$ where i goes from 1 to n
IT's just the joint distribution function of n i.i.d. copies from that distribution

It is what it's called, it is the logarithm (any base is fine) of the likelihood function
LOGs turn products into sums.

3. ## Using neyman pearson lemma

The question was to use the neyman-pearson lemma for t_0>t_1

So what I've done so far..No idea how wrong it is..

L(t)= sumof(x_i)*t^(sumof(x_i))*exp(-sumof(tx)) but we can get rid of terms without t.

So L(t_0)/L(t_1)=
(t_0)^sumof(x_i)*exp(-t_0(sumof(x_i)))/
(t_1)^sumof(x_i)*exp(-t_1(sumof(x_i)))

=(t_0)^sumof(x_i)/(t_1)^sumof(x_i) * exp(-t_0(sumof(x_i))+t_1(sumof(x_i)))

and this is <= K

Since t_0>t_1, t_1-t_0<1 and t_o/t_1 <1

so the most powerful test has critical region of the form sumof(x_i)>=K

Does this look right?

Once again sorry for the mess without the correct symbols.