To find the mean of a probability distribution, we need to sum the products of each value and its associated probability. Let's walk through the process step by step:
1. List down the hours studied and their corresponding probabilities:
[tex]\[
\begin{array}{c|c}
\text{Hours Studied} & \text{Probability} \\
\hline
0.5 & 0.07 \\
1 & 0.20 \\
1.5 & 0.46 \\
2 & 0.20 \\
2.5 & 0.07 \\
\end{array}
\][/tex]
2. Multiply each hours studied value by its corresponding probability:
[tex]\[
\begin{align*}
0.5 \times 0.07 & = 0.035 \\
1 \times 0.20 & = 0.20 \\
1.5 \times 0.46 & = 0.69 \\
2 \times 0.20 & = 0.40 \\
2.5 \times 0.07 & = 0.175 \\
\end{align*}
\][/tex]
3. Add all these products together to get the mean:
[tex]\[
\begin{align*}
0.035 + 0.20 + 0.69 + 0.40 + 0.175 & = 1.5
\end{align*}
\][/tex]
4. Thus, the mean of the probability distribution is:
[tex]\[
\boxed{1.5}
\][/tex]
