The sum should equal to 1.
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(1) That the probabilities lie between 0 and 1. (2) The sum of all probabilities of the distribution sum up to 1.
A discrete probability distribution is defined over a set value (such as a value of 1 or 2 or 3, etc). A continuous probability distribution is defined over an infinite number of points (such as all values between 1 and 3, inclusive).
Your question is not clear, but I will attempt to interpret it as best I can. When you first learn about probability, you are taught to list out the possible outcomes. If all outcomes are equally probable, then the probability is easy to calculate. Probability distributions are functions which provide probabilities of events or outcomes. A probability distribution may be discrete or continuous. The range of both must cover all possible outcomes. In the discrete distribution, the sum of probabilities must add to 1 and in the continuous distribtion, the area under the curve must sum to 1. In both the discrete and continuous distributions, a range (or domain) can be described without a listing of all possible outcomes. For example, the domain of the normal distribution (a continuous distribution is minus infinity to positive infinity. The domain for the Poisson distribution (a discrete distribution) is 0 to infinity. You will learn in math that certain series can have infinite number of terms, yet have finite results. Thus, a probability distribution can have an infinite number of events and sum to 1. For a continuous distribution, the probability of an event are stated as a range, for example, the probability of a phone call is between 4 to 10 minutes is 10% or probability of a phone call greater than 10 minutes is 60%, rather than as a single event.
For a discrete variable, you add together the probabilities of all values of the random variable less than or equal to the specified number. For a continuous variable it the integral of the probability distribution function up to the specified value. Often these values may be calculated or tabulated as cumulative probability distributions.
Not sure about only two requirements. I would say all of the following:there is a finite (or countably infinite) number of mutually exclusive outcomes possible,the probability of each outcome is a number between 0 and 1,the sum of the probabilities over all possible outcomes is 1.The Poisson distribution, for example, is countably infinite.