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 $\sigma^2 = 27.25$.

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

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.