Question 1178466
You've provided probabilities for violations (x) from 0 to 5, but they're given as a list of numbers instead of a single probability for each x. Let's fix that and then calculate the mean and variance.

**1. Correcting the Probability Distribution**

We need to make sure the probabilities sum to 1. Let's assume the list of probabilities you provided corresponds to the values of P(X=0), P(X=1), P(X=2), P(X=3), P(X=4), and P(X=5) in order.

* P(X=0) = 0.41
* P(X=1) = 0.22
* P(X=2) = 0.17
* P(X=3) = 0.13
* P(X=4) = 0.05
* P(X=5) = 0.02

Let's check if they add up to 1:
0.41 + 0.22 + 0.17 + 0.13 + 0.05 + 0.02 = 1.00

Since they sum to 1, this is a valid probability distribution.

**2. Calculating the Mean (Expected Value)**

The mean (μ) of a discrete probability distribution is calculated as:

μ = Σ[x * P(x)]

Where:
* x is the number of violations
* P(x) is the probability of x violations

Let's calculate:

μ = (0 * 0.41) + (1 * 0.22) + (2 * 0.17) + (3 * 0.13) + (4 * 0.05) + (5 * 0.02)
μ = 0 + 0.22 + 0.34 + 0.39 + 0.20 + 0.10
μ = 1.25

**3. Calculating the Variance**

The variance (σ²) of a discrete probability distribution is calculated as:

σ² = Σ[(x - μ)² * P(x)]

Or, we can use the shortcut formula:

σ² = Σ[x² * P(x)] - μ²

Let's use the shortcut formula:

First, calculate Σ[x² * P(x)]:

(0² * 0.41) + (1² * 0.22) + (2² * 0.17) + (3² * 0.13) + (4² * 0.05) + (5² * 0.02)
= (0 * 0.41) + (1 * 0.22) + (4 * 0.17) + (9 * 0.13) + (16 * 0.05) + (25 * 0.02)
= 0 + 0.22 + 0.68 + 1.17 + 0.80 + 0.50
= 3.37

Now, calculate σ²:

σ² = 3.37 - (1.25)²
σ² = 3.37 - 1.5625
σ² = 1.8075

**a. Results**

* Mean (μ) = 1.25
* Variance (σ²) = 1.8075

**b. Interpretation**

* **Mean (μ = 1.25):** The mean of 1.25 violations means that, on average, we expect to see 1.25 violations per unit of observation (whatever the unit is: person, day, etc.). It represents the long-term average number of violations.
* **Variance (σ² = 1.8075):** The variance of 1.8075 measures the spread or dispersion of the number of violations around the mean. A higher variance indicates that the number of violations tends to vary more widely, while a lower variance indicates that the number of violations tends to be closer to the mean. In this case, the variance shows that there is a moderate amount of variability in the number of violations.

**In simpler terms:**

* On average, you'll see about 1.25 violations.
* The actual number of violations can vary quite a bit, as indicated by the variance.