To find the mean of the given probability distribution, we need to use the formula for the expected value [tex]\( E(X) \)[/tex] 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] represents each possible value of the random variable [tex]\( X \)[/tex], and [tex]\( P(x_i) \)[/tex] represents the probability associated with each value.
Given the probability distribution:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Hours Studied (X)} & \text{Probability (P(X))} \\
\hline
0.5 & 0.07 \\
\hline
1 & 0.20 \\
\hline
1.5 & 0.46 \\
\hline
2 & 0.20 \\
\hline
2.5 & 0.07 \\
\hline
\end{array}
\][/tex]
Step by step, we can calculate the expected value as follows:
1. Multiply the number of hours studied by its corresponding probability:
- [tex]\( 0.5 \times 0.07 = 0.035 \)[/tex]
- [tex]\( 1 \times 0.20 = 0.20 \)[/tex]
- [tex]\( 1.5 \times 0.46 = 0.69 \)[/tex]
- [tex]\( 2 \times 0.20 = 0.40 \)[/tex]
- [tex]\( 2.5 \times 0.07 = 0.175 \)[/tex]
2. Sum these products to get the mean:
- [tex]\( 0.035 + 0.20 + 0.69 + 0.40 + 0.175 = 1.5 \)[/tex]
Therefore, the mean (expected value) of the probability distribution is [tex]\( 1.5 \)[/tex] hours.
