To find the expected value (mean) of a random variable given its probability distribution, you need to use the formula for the expected value of a discrete random variable. The expected value [tex]\( \mu \)[/tex] is calculated by summing the products of each value of the random variable and its corresponding probability:
[tex]\[ \mu = \sum_{i} x_i \cdot P(x_i) \][/tex]
Given the probability distribution:
[tex]\[
\begin{array}{r|c}
x & P \\
\hline
-3 & 0.02 \\
-1 & 0.34 \\
1 & 0.41 \\
3 & 0.15 \\
5 & 0.04 \\
7 & 0.02 \\
\end{array}
\][/tex]
We need to multiply each [tex]\(x\)[/tex] value with its corresponding probability [tex]\(P\)[/tex] and then sum up all these products.
Calculations:
[tex]\[
-3 \cdot 0.02 = -0.06
\][/tex]
[tex]\[
-1 \cdot 0.34 = -0.34
\][/tex]
[tex]\[
1 \cdot 0.41 = 0.41
\][/tex]
[tex]\[
3 \cdot 0.15 = 0.45
\][/tex]
[tex]\[
5 \cdot 0.04 = 0.20
\][/tex]
[tex]\[
7 \cdot 0.02 = 0.14
\][/tex]
Next, sum these products:
[tex]\[
\begin{align*}
\mu &= -0.06 + (-0.34) + 0.41 + 0.45 + 0.20 + 0.14 \\
&= -0.06 - 0.34 + 0.41 + 0.45 + 0.20 + 0.14
\end{align*}
\][/tex]
Adding these together:
[tex]\[
-0.06 - 0.34 + 0.41 + 0.45 + 0.20 + 0.14 = 0.80
\][/tex]
The expected value [tex]\( \mu \)[/tex] of the random variable is:
[tex]\[
\mu = 0.7999999999999999
\][/tex]
So, [tex]\(\boxed{0.7999999999999999}\)[/tex]
