Lecture 8/ Joint and Conditional Distributions L1/2017 2018 /

Definition 3 Cumulative distribution function Any function F( . } with domain the real line and counterdomain the interval [0, J] satisfying the above three properties is defined to be a cumulative distribution/unction.
This definition allows us to use the term" cumulative distribution func~
tion" without mentioning random variable.
After defining what is meant by continuous and discrete random variables in the first two subsections of the next section, we will give another property that cumulative distribution functions possess, the property of decomposition into three parts. The cumulative distribution functions defined here are univariate; the introduction of bivariate and multivariate cumulative distribution functions will be deferred until Chap. IV.
Continuous Random Variables
Definition 7 Continuous random variable A random variable X is
x called continuous if there exists a functionfx( ) such that Fx(x) ?_(-?)^x??fx(u)du? for every real number x. The cumulative distribution function Fx( .) of a continuous random variable X is called absolutely continuous.
Definition 8 Probability density function of a continuous random variable If X is a continuous random variable, the function fx( ) In ?_(-?)^x??f(u)du?
Other names that are used instead of probability density function include
density function, continuous density function, and integrating density function.
, Note that strictly speaking the probability density function fx( ) of a random variable X is not uniquely defined. All that the definition requires is that the integral of fx( ) gives Fx(x) for every x, and more than one function fx( ) may satisfy such requirement.
considerations and for this reason we will usually allow ourselves the liberty of