My Math Forum  

Go Back   My Math Forum > College Math Forum > Advanced Statistics

Advanced Statistics Advanced Probability and Statistics Math Forum

Thanks Tree1Thanks
  • 1 Post By romsek
LinkBack Thread Tools Display Modes
May 14th, 2017, 02:40 AM   #1
Senior Member
Joined: Feb 2015
From: london

Posts: 121
Thanks: 0

Likelihood function

Example in book
For the zero-truncated Poisson distribution, the probability mass function is given by:

$\displaystyle P(X = x) = p_x = \frac{λ^x}{ (e^λ − 1)x!} $ x = 1,2, ...

Suppose that we have data in which we have observed x, $\displaystyle n_x$ times, x ≥ 1. Thus we have $\displaystyle \sum_{x=1}^{\infty} n_x$ observations in total. The log likelihood can be written as:

$\displaystyle L(λ; n) = (\frac{λ^1}{(e^λ-1)1!})^{n_1} (\frac{λ^2}{(e^λ-1)2!})^{n_2} = \prod_{x=1}^{n} (\frac{λ^{x_i}}{ (e^λ − 1)x_i!})^{n_x}$

My Question:

If I look at other resources, it looks like the likelihood function is normally the product of the probability mass function, i.e:

L(λ; x) = \prod_{x=1}^{n} \frac{λ^{x_i}}{ (e^λ − 1)x_i!} $

for example:

Can anyone please explain why in the first example, they have added the power of $\displaystyle n_x$ in the formula
calypso is offline  
May 14th, 2017, 09:59 AM   #2
Senior Member
romsek's Avatar
Joined: Sep 2015
From: CA

Posts: 1,237
Thanks: 637

This is a mess.

a) do you mean you've observed $X=k$ occur $n_k$ times (It's standard to use i,j,k,l,m,n for discrete values, and x, y, u,v,y,z etc. for continuous ones)

If this is this case this problem is virtually identical to the previous one.

b) your first formulation of $L(\lambda, x)$ has the variable $i$ in it. That can't be correct.

I'm pretty sure this problem is just like the last one.
Thanks from 123qwerty
romsek is offline  

  My Math Forum > College Math Forum > Advanced Statistics

function, likelihood

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Likelihood function calypso Advanced Statistics 2 May 12th, 2017 10:05 AM
Log-likelihood function bleh Advanced Statistics 0 February 3rd, 2012 07:06 PM
Log Likelihood~ how to get the value of ?i and ?j ? ryusukekenji Advanced Statistics 1 April 20th, 2009 03:08 PM
Likelihood and Probability a1d0ru Algebra 3 October 15th, 2008 08:39 AM
Marginalize likelihood John_Smith Advanced Statistics 0 February 6th, 2008 04:28 AM

Copyright © 2017 My Math Forum. All rights reserved.