Binomial Distribution and Variance:

Binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, where the probability of success remains constant throughout the trials. The variance of a random variable is a measure of how much it varies from its expected value.

Formula for Variance:

The variance of a binomial distribution is given by the formula:

Var(X) = npq

Where,
n = number of trials
p = probability of success
q = probability of failure (q = 1 - p)

Explanation:

The variance of a binomial distribution can be derived using the following steps:

Step 1: Find the expected value (mean) of the distribution.

The expected value of a binomial distribution is given by the formula:

E(X) = np

Where,
n = number of trials
p = probability of success

Step 2: Find the variance of the distribution.

The variance of a distribution is a measure of how much the distribution varies from its expected value. It is given by the formula:

Var(X) = E(X2) - [E(X)]2

Where,
E(X2) = expected value of X2
[E(X)]2 = square of the expected value of X

To find E(X2), we use the following formula:

E(X2) = Σx2P(X = x)

Where,
Σ = sum of all values of x
P(X = x) = probability of getting x successes in n trials

For a binomial distribution, P(X = x) is given by the formula:

P(X = x) = nCxpxq(n-x)

Where,
nCx = number of ways of selecting x items from n items
px = probability of getting x successes
q(n-x) = probability of getting (n-x) failures

Using the above formulas, we can simplify the expression for E(X2) as:

E(X2) = Σx2nCxpxq(n-x)

To calculate this sum, we can use the formula:

Σx2nCx = n(n-1)p2 + np

Substituting this value in the above equation, we get:

E(X2) = n(n-1)p2q + np

Now, we can substitute the values of E(X) and E(X2) in the formula for variance:

Var(X) = E(X2) - [E(X)]2
= [n(n-1)p2q + np] - (np)2
= npq

Therefore, the variance of a binomial distribution is given by npq.

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