Correct Answer - Option 3 : E(X
2) - [E(X)]
2
Explanation:
If X is random variable assuming values x1, x2, ...., xn, with probabilities p1, p2, ....,pn respectively then the expectation of X is denoted by summation of products of assumed variable value nad there corresponding probability.
We also know that expectation of X is equal to mean (μ) of X.
∴ The expectation of X and X2 is
\(μ =E\left( X \right) = \mathop \sum \limits_{i = 1}^n {x_i}{p_i}\)
\(E\left( X^2 \right) = \mathop \sum \limits_{i = 1}^n {x^2_i}{p_i}\)
The variance of X (σ) when it is discrete is given by the addition of square of the mean of X (μ) and the expectation of X2.
\(σ^2 =\mathop \sum \limits_{i = 1}^n ({x_i-μ})^2{p_i}=\mathop \sum \limits_{i = 1}^n {x_i}^2{p_i-μ^2}\) or σ = E (X - μ)2
The standard deviation of the random variable X is defined as
\(\sigma^2=\sqrt{variance~(X)}\)
\(variance~ (X)= \mathop \sum \limits_{i = 1}^n {x^2_i}{p_i}+[{\mathop \sum \limits_{i = 1}^n {x_i}{p_i}}]^2\)
\(variance~ (X)= E(X^2)+[E(X)]^2\)