What do you notice about the probability values of random variable in each probability distribution?

Question: What do you notice about the probability values of random variable in each probability distribution?

The probability values of a random variable in each probability distribution must always satisfy certain properties, which are fundamental to probability theory. Specifically, the probability values for a random variable must be non-negative, sum to 1, and be consistent with the underlying distribution function.

In a discrete probability distribution, the probability values for each possible value of the random variable must be non-negative, meaning that they cannot be negative. They must also sum to 1, reflecting the fact that the random variable must take on one of the possible values with a total probability of 1. Additionally, the probability values must be consistent with the probability mass function, which specifies the probability of each possible value.

In a continuous probability distribution, the probability values cannot be negative and must also be consistent with the underlying probability density function, which specifies the relative likelihood of different values of the random variable. However, unlike in a discrete distribution, the probability values cannot be directly interpreted as probabilities, but instead represent the area under the probability density function over a certain range of values.

In either case, the probability values for a random variable in a probability distribution are critical for determining the likelihood of different outcomes and making predictions about future events. Ensuring that these values satisfy the necessary properties of probability theory is essential for performing accurate statistical analyses and drawing valid conclusions.