Answer :
To calculate the standard deviation \(\sigma\) of \(X\) for the given probability distribution, follow these steps:
1. Given Data:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & -5 & -1 & 0 & 2 & 5 & 10 \\ \hline P(X=x) & 0.1 & 0.1 & 0.3 & 0.1 & 0.4 & 0 \\ \hline \end{array} \][/tex]
2. Calculate the Mean (Expected Value) \(\mu\):
[tex]\[ \mu = E[X] = \sum_{i} x_i P(X = x_i) \][/tex]
[tex]\[ \mu = (-5 \times 0.1) + (-1 \times 0.1) + (0 \times 0.3) + (2 \times 0.1) + (5 \times 0.4) + (10 \times 0) \][/tex]
[tex]\[ \mu = -0.5 + (-0.1) + 0 + 0.2 + 2 + 0 = 1.6 \][/tex]
3. Calculate the Variance \(\sigma^2\):
[tex]\[ \sigma^2 = E[(X - \mu)^2] = \sum_{i} (x_i - \mu)^2 P(X = x_i) \][/tex]
[tex]\[ \sigma^2 = ((-5 - 1.6)^2 \times 0.1) + ((-1 - 1.6)^2 \times 0.1) + ((0 - 1.6)^2 \times 0.3) + ((2 - 1.6)^2 \times 0.1) + ((5 - 1.6)^2 \times 0.4) + ((10 - 1.6)^2 \times 0) \][/tex]
[tex]\[ \sigma^2 = (43.56 \times 0.1) + (7.84 \times 0.1) + (2.56 \times 0.3) + (0.16 \times 0.1) + (11.56 \times 0.4) + (70.56 \times 0) \][/tex]
[tex]\[ \sigma^2 = 4.356 + 0.784 + 0.768 + 0.016 + 4.624 + 0 = 10.548 \][/tex]
4. Calculate the Standard Deviation \(\sigma\):
[tex]\[ \sigma = \sqrt{\sigma^2} \][/tex]
[tex]\[ \sigma = \sqrt{10.548} = 3.248 \][/tex]
5. Round the Standard Deviation to Two Decimal Places:
[tex]\[ \boxed{3.23} \][/tex]
So, the standard deviation [tex]\(\sigma\)[/tex] of [tex]\(X\)[/tex] for the given probability distribution is [tex]\( \boxed{3.23} \)[/tex].
1. Given Data:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & -5 & -1 & 0 & 2 & 5 & 10 \\ \hline P(X=x) & 0.1 & 0.1 & 0.3 & 0.1 & 0.4 & 0 \\ \hline \end{array} \][/tex]
2. Calculate the Mean (Expected Value) \(\mu\):
[tex]\[ \mu = E[X] = \sum_{i} x_i P(X = x_i) \][/tex]
[tex]\[ \mu = (-5 \times 0.1) + (-1 \times 0.1) + (0 \times 0.3) + (2 \times 0.1) + (5 \times 0.4) + (10 \times 0) \][/tex]
[tex]\[ \mu = -0.5 + (-0.1) + 0 + 0.2 + 2 + 0 = 1.6 \][/tex]
3. Calculate the Variance \(\sigma^2\):
[tex]\[ \sigma^2 = E[(X - \mu)^2] = \sum_{i} (x_i - \mu)^2 P(X = x_i) \][/tex]
[tex]\[ \sigma^2 = ((-5 - 1.6)^2 \times 0.1) + ((-1 - 1.6)^2 \times 0.1) + ((0 - 1.6)^2 \times 0.3) + ((2 - 1.6)^2 \times 0.1) + ((5 - 1.6)^2 \times 0.4) + ((10 - 1.6)^2 \times 0) \][/tex]
[tex]\[ \sigma^2 = (43.56 \times 0.1) + (7.84 \times 0.1) + (2.56 \times 0.3) + (0.16 \times 0.1) + (11.56 \times 0.4) + (70.56 \times 0) \][/tex]
[tex]\[ \sigma^2 = 4.356 + 0.784 + 0.768 + 0.016 + 4.624 + 0 = 10.548 \][/tex]
4. Calculate the Standard Deviation \(\sigma\):
[tex]\[ \sigma = \sqrt{\sigma^2} \][/tex]
[tex]\[ \sigma = \sqrt{10.548} = 3.248 \][/tex]
5. Round the Standard Deviation to Two Decimal Places:
[tex]\[ \boxed{3.23} \][/tex]
So, the standard deviation [tex]\(\sigma\)[/tex] of [tex]\(X\)[/tex] for the given probability distribution is [tex]\( \boxed{3.23} \)[/tex].
