To find the mean (expected value) [tex]\(E(X)\)[/tex] of the discrete random variable [tex]\(X\)[/tex] given its probability distribution, we use the formula for the expected value of a discrete random variable. The formula is:
[tex]\[ E(X) = \sum{(X_i \cdot P(X_i))} \][/tex]
where [tex]\( X_i \)[/tex] are the values of the random variable and [tex]\( P(X_i) \)[/tex] are the corresponding probabilities.
Given the values:
[tex]\[
X = [24, 26, 27, 32, 35, 39]
\][/tex]
and their respective probabilities:
[tex]\[
P(X) = [0.16, 0.09, 0.18, 0.12, 0.24, 0.21]
\][/tex]
We will compute the weighted sum of the values:
[tex]\[ E(X) = (24 \cdot 0.16) + (26 \cdot 0.09) + (27 \cdot 0.18) + (32 \cdot 0.12) + (35 \cdot 0.24) + (39 \cdot 0.21) \][/tex]
Calculating each term individually:
[tex]\[
24 \cdot 0.16 = 3.84
\][/tex]
[tex]\[
26 \cdot 0.09 = 2.34
\][/tex]
[tex]\[
27 \cdot 0.18 = 4.86
\][/tex]
[tex]\[
32 \cdot 0.12 = 3.84
\][/tex]
[tex]\[
35 \cdot 0.24 = 8.40
\][/tex]
[tex]\[
39 \cdot 0.21 = 8.19
\][/tex]
Now, sum these values:
[tex]\[
E(X) = 3.84 + 2.34 + 4.86 + 3.84 + 8.40 + 8.19
\][/tex]
[tex]\[
E(X) = 31.47
\][/tex]
Therefore, the mean (expected value) [tex]\( E(X) \)[/tex] of this discrete random variable is:
[tex]\[ \boxed{31.47} \][/tex]
The correct answer is [tex]\( C. 31.47 \)[/tex].
