Question

If X is a random variable with Mean μ, then the variance of X, denoted by Var (X) is given by:
1. Var (X) = E[X] - μ²
2. Var (X) = E[X²] - μ
3. Var (X) = E[(X - μ)]²
4. Var (X) = E[(X - μ)]

Answer 1

Correct Answer - Option 3 : Var (X) = E[(X - μ)]²

Explanation:

If X is a random variable with mean μ, then the variance of X, denoted by Var (X) is given by:

Var (X) = E[(X - μ)]², where μ = E(X)

For a discrete random variable X, the variance of X is obtained as follows:

\(Var (X) = \sum (x-μ )^{2}pX(x)\)

where the sum is taken all over the values of x for which pX(x) > 0. So the variance of X is the weighted average of the squared deviations from the mean μ, where the weights are given by the probability function pX(x) of X.

• The standard deviation of X is defined as the square root of the variance.
• The variance cannot be negative, because it is an average of squared quantities.

• Var (X) is often denoted as σ².

Hence, If X is a random variable with mean μ, then the variance of X, denoted by Var (X) is given by Var (X) = E[(X - μ)]2