For Continous Random Variable you ARE giving the Cumulative Distribution Function, not the Probability Density Function...eh?
The cdf is defined for all random variables, discrete or continuous, so it is a better starting point than either the probability function or the density function. In one case you use differences to get the probability function and in the other you use the derivative. Most students are introduced to the derivative before the integral, so this approach is a bit more accessible --
DickBeldin
The probability that a continuous random variable X
takes a value less than or equal to x is denoted
Pr(X<=x). The probability density function of X, where X is a continuous random variable, is the
function f such that
- Pr(a<=X<=b)= INTEGRAL ( as x ranges from a thru b) f(x) dx.
Correct, but F[b]-F[a] gives the probability of an interval directly without all the complications. We hide the complications in the cdf. It is inconvenient that we can't feature the explicit form of the cdf for many of the distributions we like to use, but it is important to build the concepts with proper spacing of the difficulties. One hurdle, then a straight stretch, then a curve, then another straight ... --
DickBeldin
You may present this material as you feel best. I don't disagree with your argument. But,
mislabelling definitions is never okay. You have defined the probability density function for continuous random variables with the cumulative distribution function for the same.
RoseParks
The definitions given on this page seem much too limited. A probability distribution can be defined for random variables whose domain is not even ordered (take the multinomial, for instance). In these cases, the cumulative distribution function makes no sense. To claim, as this page does, that the distribution must have the reals as the domain is nonsense.
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