To calculate the expected value \(E(X)\) of a discrete random variable \(X\), we use the formula:
[tex]\[
E(X) = \sum_{i} x_i \cdot P(X = x_i),
\][/tex]
where \(x_i\) are the possible values of \(X\) and \(P(X = x_i)\) are the corresponding probabilities.
Given the probability distribution:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -20 & -10 & 0 & 10 & 20 & 40 \\
\hline
P(X = x) & 0.1 & 0.2 & 0.1 & 0.1 & 0 & 0.5 \\
\hline
\end{array}
\][/tex]
Let's calculate the expected value step-by-step:
1. Multiply each value of \(x\) by its corresponding probability:
[tex]\[
\begin{align*}
-20 \times 0.1 &= -2.0, \\
-10 \times 0.2 &= -2.0, \\
0 \times 0.1 &= 0.0, \\
10 \times 0.1 &= 1.0, \\
20 \times 0.0 &= 0.0, \\
40 \times 0.5 &= 20.0.
\end{align*}
\][/tex]
2. Sum these products:
[tex]\[
-2.0 + (-2.0) + 0.0 + 1.0 + 0.0 + 20.0 = 17.0.
\][/tex]
Therefore, the expected value [tex]\(E(X)\)[/tex] is [tex]\(17.0\)[/tex].
