CONTINUOUS R V continuous r v MUST be

CONTINUOUS R. V • continuous r. v MUST be defined over an interval of non-zero length [mathematics dealing with measure theory/integration] • CDF of a CONTINUOUS r. v X is itself CONTINUOUS i. e. , • This is NOT true for discrete r. v : FX has jumps when P[X=x] is non-zero. . Chpt. 4 1

CDF and PDF • For a cont. r. v, the probability distribution function (pdf) • Chpt. 4 2

Connection between P[. ] and f. X • P[x < X < x+dx] = f. X(x) dx – as well as its integral form • E[X] in terms of pdf E[g(X)] in terms of pdf f. X • Variance : Var[X] = E[X 2] - m. X 2 Chpt. 4 3

Some important pdf’s & cdf’s • • uniform r. v over interval [a, b) exponential r. v - parameter a Erlang r. v - parameters n , l Gaussian - parameters m , s Cauchy - parameters a , b Gamma - parameters a , b Laplace - parameters a , b Rayleigh - parameter a Chpt. 4 4

Guassian r. v • Gaussian - parameters m , s E[X] = m , Var[X] = s 2 • Standard Normal r. v : m = 0 , s = 1 - Standard Normal cdf : F(z) is tabulated • connection with general cdf with m , s : Chpt. 4 5

Mixed r. v • X is a mixed r. v f. X contains d-functions as well as nonzero finite values. – discrete r. v : f. X contains ONLY d-functions Chpt. 4 6

Pdf’s of derived r. v • Given r. v X and its pdf f. X(x), and the transformation to a new r. v Y by Y=g(X) then we determine the pdf f. Y(y) by – determine the CFD FY(y) =P[Y < y] – determine the PDF f. Y(y) =d FY(y)/dy Chpt. 4 7

Conditional PDF • The conditional PDF of X given a subset B of observations f. X|B(x) = f. X(x)/P[B] , x in B = 0 , otherwise • Compute conditional expectation values. . . Chpt. 4 8