Let's break down the solution step-by-step:
1. Identify the components involved in the expression [tex]$\sqrt{\Sigma\left[x^2 \cdot P(x)\right]-\mu^2}$[/tex]:
- [tex]$\Sigma\left[x^2 \cdot P(x)\right]$[/tex]: This is the sum of the squares of [tex]$x$[/tex] weighted by their probabilities [tex]$P(x)$[/tex], often referred to as the mean of the squares.
- [tex]$\mu$[/tex]: This is the mean of [tex]$x$[/tex].
2. Given values:
- The mean of the squares, [tex]$\Sigma\left[x^2 \cdot P(x)\right]$[/tex], is provided as 3.249.
- The mean, [tex]$\mu$[/tex], is provided as 1.457.
3. Calculate [tex]$\mu^2$[/tex]:
[tex]\[
\mu^2 = (1.457)^2 = 1.457 \times 1.457 = 2.122449
\][/tex]
4. Compute the difference:
[tex]\[
\Sigma\left[x^2 \cdot P(x)\right] - \mu^2 = 3.249 - 2.122449 = 1.1261509999999997
\][/tex]
5. Find the square root of the result obtained in the previous step:
[tex]\[
\sqrt{1.1261509999999997} = 1.061202619672605
\][/tex]
Therefore, the final answer is:
[tex]\[
\sqrt{\Sigma\left[x^2 \cdot P(x)\right]-\mu^2} = \sqrt{3.249 - 1.457^2} = 1.061202619672605
\][/tex]
