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

The correct option is Normal.

Let's break down the mathematical reasoning step-by-step to understand why the sum of independent and identically distributed (i.i.d.) random variables approaches a normal distribution as the sample size grows large.

1. Understanding the Setup
We are given a binomial random variable X. We have a set of n random variables, _{X1, X2, ..., Xn}, which are independent and identically distributed (i.i.d.) samples from the distribution of X. We define their sum as Y:

$$Y=\sum _{i=1}^n{X}_i$$

2. Applying the Central Limit Theorem (CLT)
The Central Limit Theorem is a fundamental theorem in probability theory. It states that, under very general conditions, if you have a sum of a large number of independent and identically distributed random variables, each with a finite mean and variance, the distribution of this sum will approximate a normal distribution as the sample size n approaches infinity ($n\to \infty$).

Here, the random variables _Xi are:
• Independent: The value of one sample does not affect the others.
• Identically Distributed: They all follow the same binomial distribution, which has a well-defined finite mean ($\mu =mp$) and finite variance (${\sigma }^2=mp(1-p)$), where m is the number of trials and p is the probability of success in the underlying binomial variable X.

3. Conclusion
Because the conditions of the Central Limit Theorem are fully satisfied, the sum Y of these n i.i.d. random variables will have a distribution that approaches a Normal distribution as $n\to \infty$.