To find the mean of the given probability distribution, we need to use the formula for the expected value (mean) of a discrete random variable. The expected value [tex]\(E(X)\)[/tex] is given by:
[tex]\[ E(X) = \sum_{i=1}^{n} x_i \cdot P(X = x_i) \][/tex]
Where [tex]\(x_i\)[/tex] represents each value of the random variable (in this case, the number of hours studied) and [tex]\(P(X = x_i)\)[/tex] is the corresponding probability.
We have the following data from the distribution table:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Hours Studied} (x_i) & \text{Probability} (P(X = x_i)) \\
\hline
0.5 & 0.07 \\
\hline
1.0 & 0.20 \\
\hline
1.5 & 0.46 \\
\hline
2.0 & 0.20 \\
\hline
2.5 & 0.07 \\
\hline
\end{array}
\][/tex]
Next, we compute the product of each value [tex]\(x_i\)[/tex] with its corresponding probability [tex]\(P(X = x_i)\)[/tex]:
1. For [tex]\(x = 0.5\)[/tex]:
[tex]\[ 0.5 \times 0.07 = 0.035 \][/tex]
2. For [tex]\(x = 1.0\)[/tex]:
[tex]\[ 1.0 \times 0.20 = 0.20 \][/tex]
3. For [tex]\(x = 1.5\)[/tex]:
[tex]\[ 1.5 \times 0.46 = 0.69 \][/tex]
4. For [tex]\(x = 2.0\)[/tex]:
[tex]\[ 2.0 \times 0.20 = 0.40 \][/tex]
5. For [tex]\(x = 2.5\)[/tex]:
[tex]\[ 2.5 \times 0.07 = 0.175 \][/tex]
Finally, we sum these products to find the expected value (mean):
[tex]\[ E(X) = 0.035 + 0.20 + 0.69 + 0.40 + 0.175 \][/tex]
[tex]\[ E(X) = 1.5 \][/tex]
So, the mean number of hours studied for this probability distribution is [tex]\(1.5\)[/tex] hours.
