# Find $E(S)$ and $Var(S)$ of a mixture of distributions [duplicate]

A class of 60 students has three sections. Summary statistics for scores on a quiz:

• Section 1: 25 students, mean $\mu_1$ = 23, SD $\sigma_1 = 3$
• Section 2: 20 students, mean $\mu_2$ = 25, SD $\sigma_2 = 2$
• Section 3: 15 students, mean $\mu_3$ = 27, SD $\sigma_3 = 4$

Let $S$ be the quiz score of a student picked at random from the class.

Found this problem in a textbook. The question was looking for something else, but I want to find $E(S)$ and $Var(S)$. How would I do that? This is what I think so far:

Let $\mathbb{I}_x$ be the event that Section $x$ is picked from the 3 sections with probability $p$, where $p$ is the $\frac{\text{number of students in that section}}{60}$ and $x \in [1,2,3]$.

Then, $$E(S) = E(E(S|\mathbb{I}_1) + E(S|\mathbb{I}_2)+ E(S|\mathbb{I}_3))$$

$$= \frac{25}{60} \cdot 23 + \frac{20}{60} \cdot 25 + \frac{15}{60} \cdot 27$$

Then, to find $Var(S)$:

$$Var(S) = (\mu^2_1 + \sigma^2_1)\frac{25}{60} +(\mu^2_2 + \sigma^2_2)\frac{20}{60} + (\mu^2_3 + \sigma^2_3)\frac{15}{60} - E(S)^2$$

$$= (23^2 + 3^2)\frac{25}{60} + (25^2 + 2^2) \frac{20}{60} + (27^2 + 4^2) \frac{15}{60} - E(S)^2$$

Does this look correct?