To find the expected value of the random variable represented by [tex]\( y \)[/tex] with the given probabilities, follow these steps:
1. List the values and their corresponding probabilities:
[tex]\[
\begin{array}{c|c}
y & \text{Probability} \\
\hline
10 & 0.10 \\
20 & 0.25 \\
30 & 0.05 \\
40 & 0.30 \\
50 & 0.20 \\
60 & 0.10 \\
\end{array}
\][/tex]
2. Calculate the contribution of each value to the expected value:
- Multiply each [tex]\( y \)[/tex] value by its corresponding probability.
[tex]\[
10 \times 0.10 = 1.0
\][/tex]
[tex]\[
20 \times 0.25 = 5.0
\][/tex]
[tex]\[
30 \times 0.05 = 1.5
\][/tex]
[tex]\[
40 \times 0.30 = 12.0
\][/tex]
[tex]\[
50 \times 0.20 = 10.0
\][/tex]
[tex]\[
60 \times 0.10 = 6.0
\][/tex]
3. Sum these contributions to get the expected value:
[tex]\[
1.0 + 5.0 + 1.5 + 12.0 + 10.0 + 6.0 = 35.5
\][/tex]
Therefore, the expected value of the random variable [tex]\( y \)[/tex] is:
[tex]\[
\boxed{35.5}
\][/tex]
