To find the mean (or expected value) of a probability distribution, we need to multiply each value (in this case, the number of hours studied) by its respective probability and then sum up all these products.
Given the probability distribution:
[tex]\[
\begin{tabular}{|c|c|}
\hline
\text{Hours Studied: } X & \text{Probability: } P(X) \\
\hline
0.5 & 0.07 \\
\hline
1.0 & 0.2 \\
\hline
1.5 & 0.46 \\
\hline
2.0 & 0.2 \\
\hline
2.5 & 0.07 \\
\hline
\end{tabular}
\][/tex]
We denote the mean of the distribution by [tex]\( \mu \)[/tex]. The mean is calculated as follows:
[tex]\[
\mu = \sum (X \times P(X))
\][/tex]
Now, let's compute it step-by-step:
1. Multiply each value of hours studied by its probability:
[tex]\[
(0.5 \times 0.07) + (1 \times 0.2) + (1.5 \times 0.46) + (2 \times 0.2) + (2.5 \times 0.07)
\][/tex]
2. Calculate each product:
[tex]\[
0.5 \times 0.07 = 0.035
\][/tex]
[tex]\[
1 \times 0.2 = 0.2
\][/tex]
[tex]\[
1.5 \times 0.46 = 0.69
\][/tex]
[tex]\[
2 \times 0.2 = 0.4
\][/tex]
[tex]\[
2.5 \times 0.07 = 0.175
\][/tex]
3. Sum all the products:
[tex]\[
0.035 + 0.2 + 0.69 + 0.4 + 0.175
\][/tex]
4. Perform the addition:
[tex]\[
0.035 + 0.2 = 0.235
\][/tex]
[tex]\[
0.235 + 0.69 = 0.925
\][/tex]
[tex]\[
0.925 + 0.4 = 1.325
\][/tex]
[tex]\[
1.325 + 0.175 = 1.5
\][/tex]
Therefore, the mean number of hours studied is:
[tex]\[
\mu = 1.5
\][/tex]
