Question

A) estimate the error in the values of the gaussian approximation of the binomial coefficients g(12,2s) as 2s changes from 0 to its maximum value. (N=12 2s between states)

B) How will the error in the value g(N,0) calculated using the gausian approximation in A if you use N=20?

Answer 1

Before getting into the details of the solution ,we need ot understand as to what exactly is a gaussian function.

A gausssian distribution is nothing but the a distribution of hte form

a for arbitarary constants a,b and C.

This can also be represented as the probability distribution function of a normally distributed function with expected value ( =variance)

or in other words,

-(i)

Now ot the question

The binomial distribution is given by (12,2s) . clearly,

n=12 and p=2s .

Thus q=1-p=1-2s (for a binomial distirbution)

In order to estimate the normal distirbution from the binomial distirbution we need to apply the Berry Esseen theorum .

The Berry-Esséen theorem gives an upper bound on the error of the distribution. It says the error is uniformly bounded by where C is a constant less than 0.7655,

ρ = and

σ is the standard deviation of Xi.

Now we can easily see that

ρ =

and σ = √(pq) f

Therefore we see that

the error in the normal (/Gaussian) approximation to a binomial(n, p) random variable is bounded by

Now observe the above expression ofr error.

You can see that the term

which drive the error value in the error formula will be maximumwhen p=q=1/2 (for a given n)

However please note that

the function   is unbounded as p approaches either 0 or 1.

What happen when 0<p<1/2

For 0 < p < 1/2, one can show that < 1/√p.

Therefore the approximation error is bounded by a constant times 1/√(np), hence the suggestion that np should be “large.”

So a conservative estimate on the error is 0.7655/√(np) when p < 1/2.

For n=12 and 0<2s<1/2

we have

The approximated error

its maximum will be

=0.312 (max )

Similarly for p=2s=0

we have

error =undefined

Thus the error in approximation of anormal distirbution from a binomial distirbution is boundeed between 0 to 0.312

b) Clearly as shown above as p tends to 0, t,the error will be undefined. regardless of finite values of n.

Thus we assume that as p tends to 0, the error tends to

the general ofrm of error is given from The Berry-Esséen theorem as

Error =