1. **Stating the problem:** You have recorded the number of hours spent on various activities in a day and want to construct a probability distribution from this data. Then, you need to compute the mean ($\mu$), variance ($\sigma^2$), and standard deviation ($\sigma$) of that distribution. 2. **Constructing the probability distribution:** - Let the total hours in a day be $T = 24$. - Suppose the hours spent on each activity are $h_1, h_2, \ldots, h_n$. - The probability of each activity is $p_i = \frac{h_i}{T}$. - These probabilities must satisfy $\sum_{i=1}^n p_i = 1$. 3. **Formulas:** - Mean (expected value): $$\mu = \sum_{i=1}^n x_i p_i$$ where $x_i$ is the value associated with the $i$th activity (in this case, the hours $h_i$ themselves). - Variance: $$\sigma^2 = \sum_{i=1}^n p_i (x_i - \mu)^2$$ - Standard deviation: $$\sigma = \sqrt{\sigma^2}$$ 4. **Step-by-step calculation:** - Calculate total hours $T = 24$. - Calculate each probability $p_i = \frac{h_i}{24}$. - Calculate mean: $$\mu = \sum_{i=1}^n h_i \times \frac{h_i}{24} = \frac{1}{24} \sum_{i=1}^n h_i^2$$ - Calculate variance: $$\sigma^2 = \sum_{i=1}^n \frac{h_i}{24} (h_i - \mu)^2$$ - Calculate standard deviation: $$\sigma = \sqrt{\sigma^2}$$ 5. **Interpretation:** - The mean $\mu$ gives the average number of hours weighted by the probability distribution. - The variance $\sigma^2$ measures how spread out the hours are from the mean. - The standard deviation $\sigma$ is the square root of variance and gives the average deviation from the mean in the same units (hours). **Note:** To complete the calculations, you need to provide the actual hours spent on each activity.