To find the expected value of the given discrete probability distribution, we follow these steps:
1. List the values and their corresponding probabilities:
[tex]\[
\begin{array}{c|cccc}
x & 1 & 4 & 7 & 10 \\
\hline
P(x) & 0.2 & 0.2 & 0.2 & 0.4 \\
\end{array}
\][/tex]
2. Multiply each value [tex]\( x \)[/tex] by its corresponding probability [tex]\( P(x) \)[/tex]:
- For [tex]\( x = 1 \)[/tex], [tex]\( P(x) = 0.2 \)[/tex]:
[tex]\[
1 \times 0.2 = 0.2
\][/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( P(x) = 0.2 \)[/tex]:
[tex]\[
4 \times 0.2 = 0.8
\][/tex]
- For [tex]\( x = 7 \)[/tex], [tex]\( P(x) = 0.2 \)[/tex]:
[tex]\[
7 \times 0.2 = 1.4
\][/tex]
- For [tex]\( x = 10 \)[/tex], [tex]\( P(x) = 0.4 \)[/tex]:
[tex]\[
10 \times 0.4 = 4.0
\][/tex]
3. Sum up all these products to get the expected value:
[tex]\[
0.2 + 0.8 + 1.4 + 4.0 = 6.4
\][/tex]
Therefore, the expected value of this probability distribution is:
[tex]\[
\boxed{6.4}
\][/tex]
