To calculate the mean of the probability distribution, we use the formula for the expected value of a discrete random variable [tex]\(X\)[/tex]:
[tex]\[ E(X) = \sum_{i} x_i \cdot P(x_i) \][/tex]
Where:
- [tex]\( x_i \)[/tex] are the values of the random variable [tex]\(X\)[/tex] (hours studied in this case).
- [tex]\( P(x_i) \)[/tex] are the corresponding probabilities for each value [tex]\( x_i \)[/tex].
Given the probability distribution:
[tex]\[
\begin{tabular}{|c|c|}
\hline
\multicolumn{2}{|c|}{Probability Distribution} \\
\hline
\begin{tabular}{c}
Hours Studied: $X$
\end{tabular} & Probability: $P(X)$ \\
\hline
0.5 & 0.07 \\
\hline
1 & 0.2 \\
\hline
1.5 & 0.46 \\
\hline
2 & 0.2 \\
\hline
2.5 & 0.07 \\
\hline
\end{tabular}
\][/tex]
We can find the mean of this distribution by calculating the weighted average of these values.
Step-by-step, the calculation is as follows:
1. Multiply each value of [tex]\(X\)[/tex] by its corresponding probability:
[tex]\[
\begin{align*}
0.5 \times 0.07 &= 0.035 \\
1 \times 0.2 &= 0.2 \\
1.5 \times 0.46 &= 0.69 \\
2 \times 0.2 &= 0.4 \\
2.5 \times 0.07 &= 0.175 \\
\end{align*}
\][/tex]
2. Add these products together to get the expected value [tex]\(E(X)\)[/tex]:
[tex]\[
0.035 + 0.2 + 0.69 + 0.4 + 0.175 = 1.5
\][/tex]
Therefore, the mean of the probability distribution is:
[tex]\[ \mathbf{1.5} \][/tex]
