GATE CS 2011 Q.18 (CS Q.18): Probability and Statistics

If the difference between the expectation of the square of a random variable ( $E[X^2]$ ) and the square of the expectation of the random variable ( $(E[X])^2$ ) is denoted by $R$ , then

$R = 0$

$R < 0$

$R \ge 0$

$R > 0$

Official answer and solution

Official answer: C
From: Official GATE 2011 answer key

Solution

Relate the given expression to the statistical variance of a random variable.

By definition, the variance of a random variable $X$ is:

\text{Var}(X) = E\left[(X - E[X])^2\right] = E[X^2] - (E[X])^2

Because $(X - E[X])^2 \ge 0$ for all real outcomes, the expectation of a non-negative quantity is always non-negative: $\text{Var}(X) \ge 0$ .

Therefore, $R = E[X^2] - (E[X])^2 = \text{Var}(X) \ge 0$ , with equality $R = 0$ holding when $X$ is a degenerate constant random variable.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2011 Computer Science and Information Technology (CS)
Question: 18 of 65, printed as CS Q.18
Section: Computer Science and Information Technology

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