of integers Z={0,±1,±2,…} Definition 10: A random variable X is called continuous if its distribution function F()is continuous for all E R and there exists a non-negative function f()on the real line such that F(x)=/f(u)da,vx∈R In defining the concept of a continuous r v. we introduced the function f(a) which is directly related to F(a) Definition 11 Let F(r) be the dF of the r v. X. The non-negative function f(ar) defined by F( f(udu, V E R--continuous F(x)=∑f(u),wr∈R-- discrete said to be the probability density function(pdf)of X Ex Let X be uniformly distributed in the interval [ a, b and we write X N U(a, b) The df of x takes the form 0 e<a r) ≤x<b, >b The corresponding pdf of X is given by <r< b ∫(x) 0 elsewhere Although we can use the distribution function F(a)as the fundamental con- cept of our probability model we prefer to adopt the density function f(a)instead,of integers Z = {0, ±1, ±2, ...}. Definition 10: A random variable X is called continuous if its distribution function F(x) is continuous for all x ∈ R and there exists a non-negative function f(·) on the real line such that F(x) = Z x −∞ f(u)du, ∀x ∈ R. In defining the concept of a continuous r.v. we introduced the function f(x) which is directly related to F(x). Definition 11: Let F(x) be the DF of the r.v. X. The non-negative function f(x) defined by F(x) = Z x −∞ f(u)du, ∀x ∈ R − −continuous or F(x) = X u≤x f(u), ∀x ∈ R − −discrete is said to be the probability density function (pdf) of X. Example: Let X be uniformly distributed in the interval [a, b] and we write X ∼ U(a, b). The DF of X takes the form: F(x) =    0 x < a, x−a b−a a ≤ x < b, 1 x ≥ b. The corresponding pdf of X is given by f(x) =  1 b−a a ≤ x ≤ b, 0 elsewhere. Although we can use the distribution function F(x) as the fundamental con￾cept of our probability model we prefer to adopt the density function f(x) instead, 18