The expression simplifies to the inequality:
\[
\left| X \right| \leq \sqrt{E \left[ \left| X \right|^2 \right]}
\]
which is a form of the Cauchy-Schwarz inequality.
Explanation:
This derivation shows the application of the Cauchy-Schwarz inequality in the context of random variables or vectors. The inequality bounds the absolute value of the expected value of a product by the product of the square roots of the expected values of the squares, which is a fundamental result in probability and linear algebra.
Step-by-step working:
- The initial expression involves the norm of a linear transformation of a vector \(X\), expressed as:
\[
|X| \leq \left( \mathbb{E} \left[ |X|^2 \right] \right)^{1/2}
\]
- The derivation uses the fact that:
\[
\left| \mathbb{E} [X] \right| \leq \left( \mathbb{E} [|X|^2] \right)^{1/2}
\]
which is the Cauchy-Schwarz inequality applied to the expectation.
- The sum notation indicates the summation over components, with \(\lambda_i\) being weights or coefficients, and the variance terms \(\sigma_i^2(x)\) representing the variance of the components.
- The inequality ultimately bounds the absolute value of the expectation by the square root of the sum of variances divided by the variance of \(X\):
\[
\left| \mathbb{E}[X] \right| \leq \sqrt{\frac{n}{\sigma^2_{X}(X)}}
\]
Conclusion:
The key takeaway is that the magnitude of the expected value of \(X\) is bounded by the standard deviation (square root of variance), which is a core concept in probability theory and statistics.