The following table shows the probability distribution for a discrete random variable.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline[tex]$X$[/tex] & 13 & 16 & 18 & 21 & 23 & 25 & 26 & 31 \\
\hline[tex]$P(X)$[/tex] & 0.07 & 0.21 & 0.17 & 0.25 & 0.05 & 0.04 & 0.13 & 0.08 \\
\hline
\end{tabular}

The mean of the discrete random variable [tex]$X$[/tex] is 20.59. What is the variance of [tex]$X$[/tex]? Round your answer to the nearest hundredth.

A. 31.06
B. 23.18
C. 33.15
D. 27.89



Answer :

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].