Random VariablesLecture 2Spring Quarter, 2002DefinitionLet E be an experiment whose outcomes are a sample space U forwhich the probability P (e) is defined for each outcome e ∈U. Arandom variable is a function X(e) that associates a number witheach outcome e ∈U.Lecture 2 1Intervals• Every interval on R corresponds to a set of outcomes in U.• Let I ∈Rbe an interval. Then A = {e ∈U: X(e)=x, x ∈ I} isan event.• P (A) can be calculated, and P (I)=P (A)• Every interval of R is an event whose probability can be calcu-lated.• For the figure below, A = {e2,e4,e5} and P (I)=P (A)=P (e2)+P (e4)+P (e5)Lecture 2 2Probability Distribution Function• Consider the semi-infinite interval Ix= {s : s ≤ x} be the intervalto the left of x.• Let X(e) be a random variable.• Let A(x) be the event that X ∈ I(x) Then A(x)={e : X(e) ≤ x}.• The probability P (X ∈ I)=P (X ≤ x)=P (A(x)) is well definedfor every x.The probability P (X ≤ x) is a special function of x called the prob-ability distribution function.FX(x)=P (X ≤ x)Lecture 2 3Probability Distribution Functionlimx→−∞FX(x)=0limx→∞FX(x)=1P (a<x≤ b)=FX(b) − FX(a)Lecture 2 4Discrete DistributionA discrete distribution function has a finite number of discontinuities.The random variable has a nonzero probability only at the points ofdiscontinuity.The distribution function for a discrete random variable is a staircasethat increases from left to right.Lecture 2 5Continuous DistributionSuppose that FX(x) is continuous for all x. Thenlimε→0FX(x) − FX(x − ε)=0so that P (X = x)=0 for allx.The derivative is well-defined where FX(x) is continuous.dFX(x)dx= limε→0FX(x) − FX(x − ε)ε= limε→0P (x − ε<X≤ x)εThe slope of the probability distribution function is equivalent to thedensity of probability.fX(x)=dFX(x)dxLecture 2 6Continuous DistributionThe distribution function (a) for a continuous random variable and(b) its probability density function. Note that the probability densityfunction is highest where the slope of the distribution function isgreatest.Lecture 2 7Continuous DistributionfX(x)=dFX(x)dxFX(x)=x−∞fX(u)duP (a<X≤ b)=FX(b) − FX(a)=bafX(u)duThe probability P (a<X≤ b) is related to the change in heightof the distribution and to the area shown in the probability densityfunction.Lecture 2 8Mixed DistributionThe range of a mixed distribu-tion contains isolated points andpoints in a continuum. The dis-tribution function is a smoothcurve except at one or more pointswhere there are finite steps.fX(x)=c(x)+kP (X = xk)δ(x−xk)c(x)=dFX/dxwhere F (x) is continuous.Lecture 2 9Random VectorLet E be an experiment whose outcomes are a sample space U forwhich the probability P (e) is defined for each outcome e ∈U.A random vector is a functionX(e)=[X1(e),X2(e),...,Xr(e)] whereXi(e) i =1, 2,...,r are random variables defined over the space U.Lecture 2 10Joint Probability Distribution FunctionFX1X2(x1,x2)=P [(X1≤ x1) ∩ (X2≤ x21. FX1X2(−∞, −∞)=02. FX1X2(−∞,x2)=0forany x23. FX1X2(x1, −∞)=0 forany x14. FX1X2(+∞, +∞)=15. FX1X2(+∞,x2)=FX2(x2)foranyx26. FX1X2(x1, +∞)=FX1(x1)foranyx1Lecture 2 11Joint Probability Distribution FunctionThe probability that an experiment produces a pair (X1,X2)thatfalls in a rectangular region with lower left corner (a, c) and upperright corner (b, d)isP [(a<X1≤ b) ∩ (c<X2≤ d)] = FX1X2(b, d) − FX1X2(a, d) −FX1X2(b, c)+FX1X2(a, c)Lecture 2 12Joint Probability Density FunctionfX1X2(x1,x2)=∂2FX1X2(x1,x2)∂x1∂x2fU,V(u, v) ≥ 0FU,V(u, v)=u−∞v−∞fU,V(ξ, η)dξdη∞−∞fU,V(ξ, η)dξdη =1FU(u)=u−∞∞−∞fU,V(ξ, η)dξdηFV(v)=∞−∞v−∞fU,V(ξ, η)dξdηfU(u)=∞−∞fU,V(u, η)dηfV(v)=∞−∞fU,V(ξ, v)dξLecture 2 13Die Tossing ExampleMapping of the outcomes of the die tossing experiment onto pointsin a plane by a particular pair of random variables.Each outcome maps into a pair of random variables.Lecture 2