91Ó°ÊÓ

The idea of non-negativity is vital in determining if a distribution can be considered a proper probability distribution.
Each value in the probability function, denoted as \( f(x) \), represents the probability of each outcome occurring.
These values must be non-negative, meaning every probability must be greater than or equal to zero.
In simpler terms, you cannot have a negative probability.
When examining the given distribution, we see the probabilities 0.1, 0.5, 0.05, 0.25, and 0.1.
Looking at each of these, we notice they are all non-negative.
This means that the first step to verifying our distribution as a discrete probability distribution—checking non-negativity—is satisfied.

Another essential criterion for a valid probability distribution is that the sum of all probabilities must equal 1.
In probability theory, this means that one of the possible outcomes must occur. We calculate the sum of the given probabilities as follows:
$$0.1 + 0.5 + 0.05 + 0.25 + 0.1 = 1.0$$
The total is exactly 1, suggesting that the sum condition is satisfied.
Without this step, a distribution cannot be considered valid because it would imply a scenario where the total probability is either less than or more than 1, which is not possible in real-world situations.

Finally, we need to verify and conclude whether the given distribution meets all the criteria for being a discrete probability distribution.
By confirming these two main criteria—non-negativity and sum of probabilities—we ensure completeness.
In this case, the given distribution satisfies both criteria:

• All values of \( f(x) \) are non-negative.
• The sum of the probabilities equals 1.

Thus, we can confidently state that the given distribution is indeed a discrete probability distribution.
Always remember that these two checks are crucial for validation and should be performed each time you evaluate any probability distribution.