To determine the variance of a discrete random variable given its probability distribution, we follow these detailed steps:
1. Identify the Mean ([tex]$\mu$[/tex]):
The mean of the discrete random variable [tex]\( X \)[/tex] is provided as 20.59.
2. Collect the Data:
We have the values of [tex]\( X \)[/tex] and their corresponding probabilities:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
X & 13 & 16 & 18 & 21 & 23 & 25 & 26 & 31 \\
\hline
P(X) & 0.07 & 0.21 & 0.17 & 0.25 & 0.05 & 0.04 & 0.13 & 0.08 \\
\hline
\end{array}
\][/tex]
3. Calculate Each Term [tex]\( (X_i - \mu)^2 \cdot P(X_i) \)[/tex]:
We calculate [tex]\( (X_i - \mu)^2 \cdot P(X_i) \)[/tex] for each value [tex]\( X_i \)[/tex] in the table.
[tex]\[
\begin{aligned}
& (13 - 20.59)^2 \cdot 0.07 = ( -7.59)^2 \cdot 0.07 \approx 4.06 \\
& (16 - 20.59)^2 \cdot 0.21 = ( -4.59)^2 \cdot 0.21 \approx 4.42 \\
& (18 - 20.59)^2 \cdot 0.17 = ( -2.59)^2 \cdot 0.17 \approx 1.14 \\
& (21 - 20.59)^2 \cdot 0.25 = ( 0.41)^2 \cdot 0.25 \approx 0.04 \\
& (23 - 20.59)^2 \cdot 0.05 = ( 2.41)^2 \cdot 0.05 \approx 0.29 \\
& (25 - 20.59)^2 \cdot 0.04 = ( 4.41)^2 \cdot 0.04 \approx 0.78 \\
& (26 - 20.59)^2 \cdot 0.13 = ( 5.41)^2 \cdot 0.13 \approx 3.81 \\
& (31 - 20.59)^2 \cdot 0.08 = (10.41)^2 \cdot 0.08 \approx 8.68 \\
\end{aligned}
\][/tex]
4. Sum Up These Values:
[tex]\[
\text{Variance} = 4.06 + 4.42 + 1.14 + 0.04 + 0.29 + 0.78 + 3.81 + 8.68 = 23.1819
\][/tex]
5. Round to the Nearest Hundredth:
The computed variance is [tex]\( 23.1819 \)[/tex]. Rounding this to the nearest hundredth, we get [tex]\( 23.18 \)[/tex].
Therefore, the variance of [tex]\( X \)[/tex] is [tex]\( \boxed{B. 23.18} \)[/tex].
