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Let's start by addressing both questions shown in the image:

### Question 3: Compute the mean and variance of the probability distribution

Given the probability distribution:
- \( x = 5, P(x) = 0.1 \)
- \( x = 10, P(x) = 0.3 \)
- \( x = 15, P(x) = 0.2 \)
- \( x = 20, P(x) = 0.4 \)

#### **Mean (Expected Value) Formula**:
\[
\mu = \sum (x_i \cdot P(x_i))
\]

\[
\mu = (5 \cdot 0.1) + (10 \cdot 0.3) + (15 \cdot 0.2) + (20 \cdot 0.4)
\]
\[
\mu = 0.5 + 3 + 3 + 8 = 14.5
\]

#### **Variance Formula**:
\[
\sigma^2 = \sum (x_i^2 \cdot P(x_i)) - \mu^2
\]

First, compute \( \sum (x_i^2 \cdot P(x_i)) \):
\[
\sum (x_i^2 \cdot P(x_i)) = (5^2 \cdot 0.1) + (10^2 \cdot 0.3) + (15^2 \cdot 0.2) + (20^2 \cdot 0.4)
\]
\[
= (25 \cdot 0.1) + (100 \cdot 0.3) + (225 \cdot 0.2) + (400 \cdot 0.4)
\]
\[
= 2.5 + 30 + 45 + 160 = 237.5
\]

Now, subtract \( \mu^2 \):
\[
\sigma^2 = 237.5 - (14.5^2) = 237.5 - 210.25 = 27.25
\]

So, the **mean** is \( \mu = 14.5 \), and the **variance** is \( \sigma^2 = 27.25 \).

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### Question 4: Discrete and Continuous Variables

- **Discrete Random Variables**: These variables take on distinct, countable values.
- **Continuous Random Variables**: These can take on any value within a given range (infinite possible values).

#### Classification:
- **a.** The number of new accounts established by a salesperson in a year: **Discrete**
- **b.** The time between customer arrivals to a bank ATM: **Continuous**
- **c.** The number of customers in Big Nick’s barber shop: **Discrete**
- **d.** The amount of fuel in your car’s gas tank: **Continuous**
- **e.** The number of minorities on a jury: **Discrete**
- **f.** The outside temperature today: **Continuous**

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Do you want more detailed steps or have any questions about these answers?

Here are 5 related questions you might find helpful:
1. How do you compute the standard deviation from variance?
2. What are some real-world applications of expected value?
3. Can a random variable be both discrete and continuous?
4. How would you calculate the mean if the probabilities do not sum to 1?
5. What is the role of probability distributions in decision-making?

**Tip**: Always check that the sum of the probabilities in a distribution equals 1 to ensure it's valid.

3. Compute the mean and variance of the following probability distribution.

4. Which of these variables are discrete and which are continuous random variables?

a. The number of new accounts established by a salesperson in a year.
b. The time between customer arrivals to a bank ATM.
c. The number of customers in Big Nick’s barber shop.
d. The amount of fuel in your car’s gas tank.
e. The number of minorities on a jury.
f. The outside temperature today.

Learn how to compute the mean and variance of a given probability distribution and classify random variables as discrete or continuous. This detailed solution covers key statistical concepts suitable for students in Grades 10-12.

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