Consider a binomial random variable X. If X1, X2,...Xnare independent and identically distributed samples from the distribution of X with sumthen the distribution of Y as nandrarr;andinfin; can be approximated as.a)Exponentialb)Bernoullic)Binomiald)NormalCorrect answer is option 'C'. Can you explain this answer?

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Consider a binomial random variable X. If X₁, X₂,...X_n are independent and identically distributed samples from the distribution of X with sum

then the distribution of Y as n → ∞ can be approximated as.

a)
Exponential
b)
Bernoulli
c)
Binomial
d)
Normal

Correct answer is option 'C'. Can you explain this answer?

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Consider a binomial random variable X. If X1, X2,...Xnare independent ...

Binomial Distribution:

A binomial distribution is a common probability distribution that occurs in practice. It arises in the following situation:

There are n independent trials.
Each trial results in a "success" or "failure"
The probability of success in each and every trial is equal to 'p'.

If the random variable X counts the number of successes in the n trials, then X has a binomial distribution with parameters n and p.

X ~ Bin (n, p).

Properties of Binomial distribution:

If X ~ Bin (n, p), then the probability mass function of the binomial distribution is

f (x) = P (X =x) = nCr p^x(1 - p)^{n - x}

for x = 0, 1, 2, 3,...,n

Mean E (X) = μ = np.

Variance (σ²) = np(1 - p).

Note:

Theorem:

Let X₁, X₂, ..., X_m be independent random variables such that X_i has a BIn (n_i, p) distribution, for i = 1, 2, ..., m. Let

Bernoulli Distribution:

A Bernoulli experiment/trial has only two possible outcomes, e.g. success/failure, heads/tails, female/male, defective/non-defective, etc.
The outcomes are typically coded as 0 (failure) or 1 (success).

X ~ Bern (p)

P (X = 1) = 1, P (X = 0) = 1 - p, 0≤p≤1

Properties:

The probability mass function is p(x) = p^x (1 - p)^{1 - x} for x = 0, 1.
The mean is E (X) = μ = (1 × p) + 0 × (1 - p) = p
Since E (X²) = (1² × p) + 0² × (1 - p) = p,
σ² = var (X) = E (X²) - μ² = p - p² = p (1 - p).

Note:

The Bernoulli distribution is a special case of binomial distribution with n = 1.

Exponential Distribution:

The probability density function of the exponential distribution is,

Normal Distribution:

The probability density function of normal distribution is given by,

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Consider a binomial random variable X. If X1, X2,...Xnare independent and identically distributed samples from the distribution of X with sumthen the distribution of Y as n→∞ can be approximated as.a)Exponentialb)Bernoullic)Binomiald)NormalCorrect answer is option 'C'. Can you explain this answer?

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