Consider a binomial random variable X. If X₁, X₂,...X_n are independent and identically distributed samples from the distribution of X with sum \(Y = \mathop \sum \limits_{i = 1}^n {X_i}\) then the distribution of Y as n → ∞ can be approximated as.

Question

Prepp Team · Accepted Answer

Binomial Random Variable and its Sum

The question asks about the distribution of \(Y\), which is the sum of \(n\) independent and identically distributed (i.i.d.) samples \(X_1, X_2, \ldots, X_n\) drawn from a binomial random variable \(X\). We need to determine what the distribution of \(Y\) can be approximated as when \(n\) approaches infinity.

Understanding Binomial Distributions

A binomial random variable counts the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and a constant probability of success.
If a random variable \(X\) follows a binomial distribution, it is denoted as \(X \sim B(k, p)\), where \(k\) is the number of trials and \(p\) is the probability of success in each trial.

Sum of Independent Binomial Variables

A crucial property in probability theory is how certain distributions behave when their independent instances are summed. For binomial random variables, there's a specific rule:

Let's assume that each of the i.i.d. samples \(X_i\) is drawn from a binomial distribution, say \(X_i \sim B(k, p)\). This means each \(X_i\) represents the number of successes observed in \(k\) trials, with the probability of success being \(p\).
When we sum these \(n\) independent binomial random variables, \(Y = \mathop \sum \limits_{i = 1}^n {X_i}\), we are essentially combining the total number of successes from all \(n\) sets of \(k\) trials.
Since each \(X_i\) contributes \(k\) trials, the total number of trials for the sum \(Y\) becomes \(n \times k\).
The probability of success \(p\) remains constant across all these combined trials.
Therefore, \(Y\) represents the total number of successes obtained from a grand total of \(nk\) independent Bernoulli trials, each with the same probability of success \(p\).
By definition, this combined scenario also fits the criteria for a binomial distribution. So, \(Y \sim B(nk, p)\).

Approximation as n → ∞

The phrase "as \(n \to \infty\)" in the question is important. While the Central Limit Theorem (CLT) suggests that the sum of a large number of i.i.d. random variables, regardless of their underlying distribution (provided mean and variance are finite), will tend towards a normal distribution, we must also consider the exact nature of the distribution family itself.

The distribution of \(Y\) is inherently binomial. Even as \(n\) increases towards infinity, the sum \(Y\) remains a binomial random variable, just one with an increasingly large number of trials (\(nk\)).
For a binomial distribution \(B(N, p)\) where \(N\) is very large (in our case, \(N = nk\)), it can indeed be well-approximated by a normal distribution under certain conditions (typically, when \(Np > 5\) and \(N(1-p) > 5\)).
However, the question asks what the distribution of \(Y\) "can be approximated as". Since the exact distribution of the sum of independent binomial variables is binomial, it naturally "can be approximated as" a binomial distribution (albeit with adjusted parameters \(nk\)). This is a direct and fundamental property of binomial distributions under summation.

Conclusion

The sum of independent and identically distributed binomial random variables results in another binomial random variable. As \(n\) approaches infinity, the number of trials for this resulting binomial distribution (\(nk\)) also approaches infinity, but the distribution type remains binomial.