University of Alberta Guide/STAT/222/Formulas and Functions:Cumulative Distribution Functions

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The $$F_{X}(x) \rightarrow (cdf)\,$$, represents the probability that the value of a random variable $$X\,$$, regardless of its distribution, will be less than or equal to the value $$x\,$$ that is provided. For continuous random variables, $$F_{X}(0) = 0\,$$ because $$P(X = 0) = 0\,$$ for all continuous random variables. Therefore, all cumulative distribution functions will start at zero. Generally, $$F_{X}(x)\,$$ represents $$P(X \leq x)\,$$, however, if what is desired is $$P(X > x)\,$$ then $$1 - F_{X}(x)\,$$ would be used. CDF for an Exponential RV Notice that as $$x\,$$ increases, the probability nears closer and closer to one. This is because as $$x\,$$ increases, the area under the curve of the PDF becomes closer and closer to one.