Define probability distribution
f(x) = x−2/10,x=0,1,2,3,4

A **probability distribution** assigns probabilities to each possible outcome of a random variable in a sample space. To qualify as a valid probability distribution, two conditions must be met:
1. Each individual probability must be non-negative: \( f(x) \geq 0 \).
2. The total probability must sum to 1: \( \sum f(x) = 1 \).

Given the function \( f(x) = \frac{x - 2}{10} \) for \( x = 0, 1, 2, 3, 4 \), let's check if it qualifies as a valid probability distribution.

### 1. Check non-negativity condition:
Evaluate \( f(x) \) for each value of \( x \):

- \( f(0) = \frac{0 - 2}{10} = -\frac{2}{10} = -0.2 \)
- \( f(1) = \frac{1 - 2}{10} = -\frac{1}{10} = -0.1 \)
- \( f(2) = \frac{2 - 2}{10} = \frac{0}{10} = 0 \)
- \( f(3) = \frac{3 - 2}{10} = \frac{1}{10} = 0.1 \)
- \( f(4) = \frac{4 - 2}{10} = \frac{2}{10} = 0.2 \)

Since two of the values, \( f(0) \) and \( f(1) \), are negative, this function does not meet the first condition of non-negativity.

Thus, **this function is not a valid probability distribution** because it assigns negative probabilities, which is not allowed.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How can we adjust this function to create a valid probability distribution?
2. What are the characteristics of a discrete probability distribution?
3. How do we verify if a function is a probability density function in the continuous case?
4. What is the difference between a cumulative distribution function (CDF) and a probability distribution?
5. Can a probability distribution assign zero probability to some outcomes?

**Tip**: Always ensure that probabilities are non-negative and sum up to 1 when working with probability distributions.

Define probability distribution f(x) = (x − 2)/10, x = 0, 1, 2, 3, 4

Learn how to check if f(x) = (x − 2)/10 for x = 0, 1, 2, 3, 4 is a valid probability distribution. This problem involves evaluating non-negativity and summation conditions, essential for understanding probability distributions. Suitable for Grades 11-12.

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