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A Uniform Distribution is a type of probability distribution where all outcomes are equally probable. It's often visualized as a rectangle on a graph, indicating a constant likelihood over a specified range. For example, when you use a random number generator to pick a number between 0 and 100, every possible number has the same chance to be selected. This scenario perfectly illustrates a uniform distribution.
The function representing this distribution is a Probability Density Function (PDF), which is flat, signifying uniformity in probability. The PDF of a uniform distribution between 0 and 100 will thus be horizontal, reflecting that no number is given preferential probability over another.

Simply put, a Random Variable is a numerical value determined by the outcome of a random process. In statistical terms, it's a variable whose possible values are numerical outcomes of a random phenomenon. When you use a random number generator to get a real number between 0 and 100, the chosen number becomes the random variable, denoted by "x."
Random variables can be discrete (specific numbers, like roll outcomes of a die) or continuous (any value in a range, like temperature). For the uniform distribution between 0 and 100, the random variable is continuous, as any number within that range is possible.

Probability quantifies how likely an event is to happen. It ranges from 0 (impossible event) to 1 (certain event).
For our uniform distribution example, probability is evenly spread across the range from 0 to 100. You can think of the probability as the area under the PDF curve which is always 1, representing 100%. Each individual real number, however, has a negligible or infinitesimally small probability, but every number collectively makes up the total probability space.
Calculating the probability of selecting a specific number in a continuous range, like 0 to 100, involves understanding the density function instead. Here, the density function helps define the likelihood of x falling within certain intervals.

Probability Space is a mathematical construct used to model a random process, encompassing all possible outcomes and their probabilities. It consists of three main components:

• Sample Space: all possible outcomes (in our example, any number between 0 to 100).
• Event: any outcome or a set of outcomes of the process.
• Probability Function: a rule assigning probabilities to each event.

The PDF provides the probability function for continuous outcomes. In our uniform distribution example, choosing a number between 0 and 100, the probability space is defined by the range over which the PDF is constant. Every number in this range is part of the sample space, and their collective probability, represented by the area under the PDF, equals 1.