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To determine whether the table represents a probability distribution, we need to check if it satisfies the following conditions:

1. Each probability \( P(x) \) must be between 0 and 1 (inclusive).
2. The sum of all probabilities must equal 1.

### Step 1: Verify each \( P(x) \) value
We can see that each probability listed is within the range [0, 1]:
- \( P(0) = 0.025 \)
- \( P(1) = 0.148 \)
- \( P(2) = 0.335 \)
- \( P(3) = 0.355 \)
- \( P(4) = 0.137 \)

Since all probabilities are between 0 and 1, the first condition is satisfied.

### Step 2: Check the sum of the probabilities
We now add the probabilities to check if they sum to 1:

\[
\sum P(x) = 0.025 + 0.148 + 0.335 + 0.355 + 0.137 = 1.000
\]

Since the sum is exactly 1, the second condition is also satisfied.

### Conclusion
Since both conditions are satisfied, this is a valid probability distribution.

### Step 3: Calculate the mean (expected value) and standard deviation
To calculate the mean \( \mu \) and standard deviation \( \sigma \), we use the following formulas:

- **Mean \( \mu \):**

\[
\mu = \sum x P(x)
\]

- **Standard deviation \( \sigma \):**

\[
\sigma = \sqrt{\sum (x - \mu)^2 P(x)}
\]

Let me calculate these values.### Results:
- **Mean (Expected value), \( \mu \):** 2.431
- **Standard deviation, \( \sigma \):** 0.982

Thus, the probability distribution has a mean of 2.431 households with printers and a standard deviation of approximately 0.982.

Do you want more details or have any questions?

Here are 5 related questions:
1. How do you calculate the variance of a probability distribution?
2. What is the significance of the mean in a probability distribution?
3. Can we compute the median for this discrete probability distribution?
4. How do standard deviation and variance relate in probability distributions?
5. What is the probability of having at least 3 households with printers in this scenario?

**Tip:** When solving for the mean of a probability distribution, always multiply each outcome by its probability, then sum the results!

Households are randomly selected and partitioned into groups of four. For those groups, the random variable x is the number of households with a printer. Determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.

Determine whether the table represents a probability distribution for the number of households with printers. Learn how to calculate the mean and standard deviation with step-by-step guidance and ensure that the probability conditions are satisfied. Suitable for students in Grades 9-12.

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