Discussion
Let X be a random variable. Define the random variable \(Y = X - \langle X \rangle\). Let \(\langle X \rangle = \mu\) and \(\langle (X-\mu)^{2} \rangle = \sigma^{2}\). Choose the correct alternative :
Answer: B
The correct statement is \(\langle Y \rangle = 0\) and \(\langle (Y - \langle Y \rangle)^{2} \rangle = \sigma^{2}\).
Shifting a random variable by a constant shifts its mean by that constant and leaves its spread unchanged:
\[ \begin{aligned} \langle Y \rangle &= \langle X \rangle - \mu = 0 \\ \langle (Y - 0)^{2} \rangle &= \langle (X - \mu)^{2} \rangle = \sigma^{2} \end{aligned} \]
So \(Y\) is the centred version of \(X\), with the same variance. A mean of \(\mu\) would need no shift at all; a variance of 0 would mean \(Y\) is constant; and \(\sigma^{2} - \mu^{2}\) mixes up the variance formula \(\langle X^{2} \rangle - \mu^{2}\) with the variance itself.
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