Sure, let's find the expected value of the random variable, given its probability distribution.
1. Identify the values of the random variable [tex]\(x\)[/tex]:
[tex]\[
x = [-1, 1, 3, 5, 7, 9]
\][/tex]
2. Identify the corresponding probabilities [tex]\(P(x)\)[/tex]:
[tex]\[
P = [0.05, 0.18, 0.21, 0.23, 0.14, 0.19]
\][/tex]
3. Recall the formula for the expected value [tex]\(\mathbb{E}(X)\)[/tex] of a discrete random variable:
[tex]\[
\mathbb{E}(X) = \sum_{i=1}^{n} x_i \cdot P(x_i)
\][/tex]
where [tex]\(x_i\)[/tex] are the values of the random variable and [tex]\(P(x_i)\)[/tex] are the corresponding probabilities.
4. Plug in the values and their probabilities into the formula:
[tex]\[
\mathbb{E}(X) = (-1) \cdot 0.05 + 1 \cdot 0.18 + 3 \cdot 0.21 + 5 \cdot 0.23 + 7 \cdot 0.14 + 9 \cdot 0.19
\][/tex]
5. Calculate each term individually:
[tex]\[
-1 \cdot 0.05 = -0.05
\][/tex]
[tex]\[
1 \cdot 0.18 = 0.18
\][/tex]
[tex]\[
3 \cdot 0.21 = 0.63
\][/tex]
[tex]\[
5 \cdot 0.23 = 1.15
\][/tex]
[tex]\[
7 \cdot 0.14 = 0.98
\][/tex]
[tex]\[
9 \cdot 0.19 = 1.71
\][/tex]
6. Add all these terms together to find the expected value:
[tex]\[
\mathbb{E}(X) = -0.05 + 0.18 + 0.63 + 1.15 + 0.98 + 1.71 = 4.60
\][/tex]
Therefore, the expected value of the random variable is [tex]\( \boxed{4.60} \)[/tex].
