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.