To find the expected value (or mean) [tex]\(\mu\)[/tex] of the random variable with the given probability distribution, we follow these steps:
1. Identify the values of the random variable [tex]\(x\)[/tex] and their corresponding probabilities [tex]\(P\)[/tex]:
[tex]\[
\begin{array}{|r|c|}
\hline
x & P \\
\hline
-1 & 0.05 \\
1 & 0.18 \\
3 & 0.21 \\
5 & 0.23 \\
7 & 0.14 \\
9 & 0.19 \\
\hline
\end{array}
\][/tex]
2. Use the formula for the expected value of a discrete random variable:
[tex]\[
\mu = \sum (x_i \cdot P_i)
\][/tex]
where [tex]\(x_i\)[/tex] are the values of the random variable and [tex]\(P_i\)[/tex] are the corresponding probabilities.
3. Multiply each value [tex]\(x_i\)[/tex] by its corresponding probability [tex]\(P_i\)[/tex] and then sum these products:
[tex]\[
\mu = (-1 \cdot 0.05) + (1 \cdot 0.18) + (3 \cdot 0.21) + (5 \cdot 0.23) + (7 \cdot 0.14) + (9 \cdot 0.19)
\][/tex]
4. Calculate each term:
[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]
5. Sum these values up:
[tex]\[
\mu = -0.05 + 0.18 + 0.63 + 1.15 + 0.98 + 1.71
\][/tex]
6. Add the values together:
[tex]\[
\mu = 4.6000000000000005
\][/tex]
Thus, the expected value [tex]\(\mu\)[/tex] of the random variable is exactly:
[tex]\[
\mu = 4.6000000000000005
\][/tex]
