Significance of cumulative distribution function
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In probability theory and statistics, thecumulative distribution function (CDF, also cumulative density function) of a real-valued random variable X, or justdistribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
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Start with your random variable: XX. The simplest definition of a random variable requires it to take values in the real numbers, so that's the approach I'll take.
So you know that XX takes on numerical values between negative infinity and infinity. One natural question to ask is, "How likely is it that XX will not exceed 0?" An equally reasonable question is, "How likely is it that XX will not exceed 1?" Each of these questions has a numerical answer between 00 and 11(inclusive) because each is asking for the probability of an event and probabilities are required to be in this interval.
A mathematician quickly generalizes this idea to, "How likely is it that XX will not exceed ww?" where ww can be ANY real number. Once we ask this question, though, we are no longer looking for a single number to describe the probability of an event, we are now looking for a (possibly) different number for EVERY single value of ww between negative infinity and infinity.
This collection of numbers (each of which is the probability of a different event of the form {X≤w}{X≤w}) is called the cumulative probability function. It maps every real number ww to a probability between 00 and 11. Once you find this function, you can just plug in the ww that you want, and the function spits out the associated probability.
In textbooks and classrooms, the Cumulative Distribution Function is typically written as:
F(x)=P(X≤x)F(x)=P(X≤x)
In my explanation, I used ww in place of xxbecause newcomers to the field are often confused when they see a XX (i.e. the random variable) and xx (i.e. any real number) in the same expression. F(w)=P(X≤w)F(w)=P(X≤w) is equivalent to the usual textbook definition
So you know that XX takes on numerical values between negative infinity and infinity. One natural question to ask is, "How likely is it that XX will not exceed 0?" An equally reasonable question is, "How likely is it that XX will not exceed 1?" Each of these questions has a numerical answer between 00 and 11(inclusive) because each is asking for the probability of an event and probabilities are required to be in this interval.
A mathematician quickly generalizes this idea to, "How likely is it that XX will not exceed ww?" where ww can be ANY real number. Once we ask this question, though, we are no longer looking for a single number to describe the probability of an event, we are now looking for a (possibly) different number for EVERY single value of ww between negative infinity and infinity.
This collection of numbers (each of which is the probability of a different event of the form {X≤w}{X≤w}) is called the cumulative probability function. It maps every real number ww to a probability between 00 and 11. Once you find this function, you can just plug in the ww that you want, and the function spits out the associated probability.
In textbooks and classrooms, the Cumulative Distribution Function is typically written as:
F(x)=P(X≤x)F(x)=P(X≤x)
In my explanation, I used ww in place of xxbecause newcomers to the field are often confused when they see a XX (i.e. the random variable) and xx (i.e. any real number) in the same expression. F(w)=P(X≤w)F(w)=P(X≤w) is equivalent to the usual textbook definition
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