## Answer :

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]