Characteristic function of binomial distribution.

[mnb96]

Hello,
I considered a Binomial distribution B(n,p), and a discrete random variable $$X=\frac{1}{n}B(n,p)$$. I tried to compute the characteristic function of X and got the following:

$$\phi_X(\theta)=E[e^{i\frac{\theta}{n}X}]=(1-p+pe^{i\theta/n})^n$$

I tried to compute the limit for $n\to +\infty$ and I got the following result:

$$\lim_{n\to\infty}\phi_X(\theta)=e^{ip\theta}$$

How should I interpret this result?
That characteristic function would correspond to a delta-function distribution centered at -p. It doesn't make much sense to me.

 

[mathman]

Except for the sign (it should be a delta function at p) it makes sense. By the law of large numbers the average of a sequence of random binomials will converge to p.

 

[mnb96]

Ok thanks!
Now it's more clear.

If I got it right, that basically means that if I pick up an infinite amount of coins (head=1, tails=0, with probabilities p and (1-p)), throw them all at once, and finally sum up the result and divide by the number of coins, I should obtain p with probability 1.

 

[mathman]

Ok thanks!
Now it's more clear.

If I got it right, that basically means that if I pick up an infinite amount of coins (head=1, tails=0, with probabilities p and (1-p)), throw them all at once, and finally sum up the result and divide by the number of coins, I should obtain p with probability 1.

You've got the point, although mathematical precision means you talk about a limit.

 

[blue_raver22]

Could you help me understand what this means?


The characteristic function of a distribution is a mathematical tool that describes the distribution in terms of its moments. In this case, the characteristic function of X, a transformed Binomial distribution, is approaching a delta-function distribution centered at -p as n approaches infinity. This means that as n increases, the distribution becomes more and more concentrated at -p, with almost all of the probability mass located at that point. This result is expected because as n increases, the transformed Binomial distribution becomes more and more similar to a Bernoulli distribution with probability of success p, which is a delta-function distribution at p. This also highlights the importance of understanding the limits and assumptions made when using mathematical tools, such as the characteristic function, in analyzing and interpreting data.