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The X is a discrete random variable assuming values x1, x2, ...., xn, with probabilities p1, p2, ....,pn respectively, then the variance of X is given by
1. E(X2)
2. E(x2) + E(x)
3. E(X2) - [E(X)]2
4. \(\sqrt{E(X^2)-[E(X)]^2}\)

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Correct Answer - Option 3 : E(X2) - [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 Xis

 \(μ =E\left( X \right) = \mathop \sum \limits_{i = 1}^n {x_i}{p_i}\)

\(E\left( X^2 \right) = \mathop \sum \limits_{i = 1}^n {x^2_i}{p_i}\)

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)

The standard deviation of the random variable X is defined as

\(\sigma^2=\sqrt{variance~(X)}\)

\(variance~ (X)= \mathop \sum \limits_{i = 1}^n {x^2_i}{p_i}+[{\mathop \sum \limits_{i = 1}^n {x_i}{p_i}}]^2\)

\(variance~ (X)= E(X^2)+[E(X)]^2\)

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