To find the mean of the probability distribution, we will calculate the expected value of the number of hours studied. The expected value [tex]\( E(X) \)[/tex] of a discrete random variable [tex]\( X \)[/tex] is given by the formula:
[tex]\[ E(X) = \sum [X_i \cdot P(X_i)] \][/tex]
Where:
- [tex]\( X_i \)[/tex] represents the number of hours studied.
- [tex]\( P(X_i) \)[/tex] represents the probability associated with [tex]\( X_i \)[/tex].
We are given the following probabilities and corresponding hours studied:
\begin{tabular}{|c|c|}
\hline
Hours Studied ([tex]\(X_i\)[/tex]) & Probability ([tex]\(P(X_i)\)[/tex]) \\
\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}
Now we can multiply each number of hours studied by its corresponding probability and sum the results:
[tex]\[
E(X) = (0.5 \times 0.07) + (1 \times 0.2) + (1.5 \times 0.46) + (2 \times 0.2) + (2.5 \times 0.07)
\][/tex]
Calculating each term individually:
[tex]\[
0.5 \times 0.07 = 0.035 \\
1 \times 0.2 = 0.2 \\
1.5 \times 0.46 = 0.69 \\
2 \times 0.2 = 0.4 \\
2.5 \times 0.07 = 0.175
\][/tex]
Now, sum these products together to find the mean:
[tex]\[
E(X) = 0.035 + 0.2 + 0.69 + 0.4 + 0.175 = 1.5
\][/tex]
Therefore, the mean of the probability distribution is:
[tex]\[
\boxed{1.5}
\][/tex]
