To find the expected value [tex]\(\mu\)[/tex] of a random variable with a given probability distribution, we use the formula for the expected value of a discrete random variable:
[tex]\[ \mu = \sum_{i} x_i \cdot P(x_i) \][/tex]
This means we multiply each possible value [tex]\(x_i\)[/tex] of the random variable by the corresponding probability [tex]\(P(x_i)\)[/tex] and then sum up all these products. Let's do this step by step with the given values.
Given:
[tex]\[
\begin{array}{r|c}
x & P(x) \\
\hline
-1 & 0.05 \\
1 & 0.18 \\
3 & 0.21 \\
5 & 0.23 \\
7 & 0.14 \\
9 & 0.19 \\
\end{array}
\][/tex]
1. Multiply each [tex]\(x\)[/tex] value by its corresponding probability:
- For [tex]\(x = -1\)[/tex]: [tex]\(-1 \cdot 0.05 = -0.05\)[/tex]
- For [tex]\(x = 1\)[/tex]: [tex]\(1 \cdot 0.18 = 0.18\)[/tex]
- For [tex]\(x = 3\)[/tex]: [tex]\(3 \cdot 0.21 = 0.63\)[/tex]
- For [tex]\(x = 5\)[/tex]: [tex]\(5 \cdot 0.23 = 1.15\)[/tex]
- For [tex]\(x = 7\)[/tex]: [tex]\(7 \cdot 0.14 = 0.98\)[/tex]
- For [tex]\(x = 9\)[/tex]: [tex]\(9 \cdot 0.19 = 1.71\)[/tex]
2. Sum up all these products:
[tex]\[
-0.05 + 0.18 + 0.63 + 1.15 + 0.98 + 1.71
\][/tex]
3. Perform the addition:
[tex]\[
-0.05 + 0.18 = 0.13
\][/tex]
[tex]\[
0.13 + 0.63 = 0.76
\][/tex]
[tex]\[
0.76 + 1.15 = 1.91
\][/tex]
[tex]\[
1.91 + 0.98 = 2.89
\][/tex]
[tex]\[
2.89 + 1.71 = 4.60
\][/tex]
Thus, the expected value [tex]\(\mu\)[/tex] of the random variable is:
[tex]\[
\mu = 4.6000000000000005
\][/tex]
Hence, the expected value of the given random variable is:
[tex]\(\mu = 4.6000000000000005\)[/tex].
