Limit of rescaled Binomials
Let Xn be a binomial random variable with parameters n and p = lambda/n, where lambda is a fixed positive number.
Let k be a natural number. As n tends to infinity, the probability mass function P(Xn = k) converges to a number that only depends on lambda and k. What is the limit?
To find the limit of P(Xn = k) as n tends to infinity, we can use the concept of the Poisson distribution.
The probability mass function of a binomial random variable Xn with parameters n and p can be written as:
P(Xn = k) = nCk * p^k * (1-p)^(n-k)
Where nCk represents the binomial coefficient.
Now, let's substitute p = lambda/n into the equation:
P(Xn = k) = nCk * (lambda/n)^k * (1-(lambda/n))^(n-k)
Simplifying the expression:
P(Xn = k) = nCk * (lambda/n)^k * ((n-lambda)/n)^(n-k)
As n tends to infinity, the binomial coefficient nCk can be approximated by the binomial coefficient of the limiting distribution, which is the Poisson distribution with parameter lambda. We can write:
nCk ≈ lambda^k / k!
Also, observe that ((n-lambda)/n)^(n-k) tends to 1 as n tends to infinity.
Substituting these approximations into the equation:
P(Xn = k) ≈ (lambda^k / k!) * (lambda/n)^k * 1
= (lambda^k / k!) * (lambda^k / n^k)
= (lambda^2k / (k!) * n^k)
As n tends to infinity, the term (lambda^2k / (k!) * n^k) tends to 0 as the denominator n^k increases faster than the numerator lambda^2k.
Therefore, the limit of P(Xn = k) as n tends to infinity is 0 for k ≠ 0, and when k = 0, it converges to (lambda^0 / 0!) = 1.
In summary, the limit of the probability mass function P(Xn = k) as n tends to infinity is:
1 for k = 0
0 for k ≠ 0
The limiting distribution is a degenerate Poisson distribution with parameter lambda, where the probability of observing a non-zero value is 0.