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Continuous Random Variables & Distributions

A continuous random variable X is one which can take on any real value in some range of (or anywhere along) the real line (-¥ ,+¥ ). Unlike with discrete r.v.’s, the measurement scale for a continuous r.v. can be subdivided to any extent desired and its value could lie in any of these subdivisions.

Since a continuous r.v. can take on infinitely many different values, it is impossible to associate a probability with each value (unlike with discrete r.v.’s). Thus, a p.m.f cannot be defined for a continuous r.v.

Instead we define a probability density function(or pdf) for X as a function f(x) such that give two real numbers a and b with a£ b,

Note that if we plot f(x) as a function of x, then the probability above is the area between a and b under the curve representing this function :

A valid pdf must satisfy:

    1. f(x) ³ 0 for all x
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Next: Continuous Random Variables: Percentiles and the c.d.f.
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Jayant Rajgopal