536 Chapter 9. Essential Statistics for Data Analysis
quotinghand hmight not be sufficient and one should present the distribution
function plot as well.
Up until now we have assumed that the likelihood function is described by a
single variableh.Ifwehaveknumber of parameters instead, we will have to solve
the followingksimultaneous equations to find the maximum likelihood solution.
∂lnL(h 1 ,h 2 , ..., hk)
∂hi∣
∣
∣
hi=h∗i= 0 (9.3.23)
Now that we know what maximum likelihood function is, what do we do with it?
Well, to do any Maximum likelihood analysis we first need a probability distribution
function. In the next section we will look at some commonly used distribution
functions and employ the Maximum likelihood methodology to draw inferences about
them.
9.3.D SomeCommonDistributionFunctions.............
D.1 BinomialDistribution
The binomial distribution can be used to determine the probability ofrsuccesses
out ofNoutcomes of an experiment and is defined by
f(r;N, p)=N!
r!(N−r)!pr(1−p)N−r, (9.3.24)withr=0, 1 , ..., Nand 0≤p≤1.
The events must be random, mutually exclusive, and independent, which in sim-
ple terms essentially means that the occurrence of one event should not influence
the outcome of the next. The outcome of an experiment describable by binomial
distribution has only two possible outcomes, such as getting head or tails when a
coin is flipped or detecting or failing to detect a particle when it passes through the
active medium of a detector. This means that if the probability of getting an event
ispthen probability of not seeing the event would simply be (1−p).
Let us now write the likelihood function for the occurrence of an event and then
try to calculate its most probable value and the corresponding error. The likelihood
function for a continuous variablepthat follows binomial distribution can be written
as
L(p)=N!
r!(N−r)!pr(1−p)N−r. (9.3.25)To compute the most probable valuep∗ofp, we take the derivative of its natural
logarithm with respect topand then equate it to zero (see equation 9.3.21). First
we take the logarithm of the function keeping in view that we are only interested in
evaluating terms that explicitly containp.
lnL(p)=ln[
N!
r!(N−r)!pr(1−p)N−r]
= rln(p)+(N−r)ln(1−p)+ln[
N!
r!(N−r)!]
(9.3.26)
The derivative of this with respect topis
∂ln(L)
∂p=
r
p−
N−r
1 −p