REGRESSION MODELS
THE CDF AND PDF OF THE STANDARD NORMAL DISTRIBUTION PROBABILITIES CAN BE COMPUTED IN TERMS OF THE PDF AS WELL AS THE CDF NOTE THAT, BY (1 03) AND THE FUNDAMENTAL THEOREM OF CALCULUS ONCE MORE, PR(A ?X ?B) = F(B) ?F(A) = Z B A F(X) DX (1 05) SINCE (1 05) MUST HOLD FOR ARBITRARY A AND B, IT IS CLEAR WHY F(X) MUST ALWAYS BE NONNEGATIVE HOWEVER, IT IS IMPORTANT TO REMEMBER THAT F(X) IS NOT BOUNDED ABOVE BY UNITY, BECAUSE THE VALUE OF A PDF AT A POINT X IS NOT A PROBABILITY ONLY WHEN A PDF IS INTEGRATED OVER SOME INTERVAL, AS IN (1 05), DOES IT YIELD A PROBABILITY THE MOST COMMON EXAMPLE OF A CONTINUOUS DISTRIBUTION IS PROVIDED BY THE NORMAL DISTRIBUTION THIS IS THE DISTRIBUTION THAT GENERATES THE FAMOUS OR INFAMOUS “BELL CURVE” SOMETIMES THOUGHT TO IN?UENCE STUDENTS’ GRADE DISTRI- BUTIONS THE FUNDAMENTAL MEMBER OF THE NORMAL FAMILY OF DISTRIBUTIONS IS THE STANDARD NORMAL DISTRIBUTION IT IS A CONTINUOUS SCALAR DISTRIBUTION, DE?NED REGRESSION MODELS F(X) X P FIGURE 1 2 THE CDF OF A BINARY RANDOM VARIABLE ON THE ENTIRE REAL LINE THE PDF OF THE STANDARD NORMAL DISTRIBUTION IS OFTEN DENOTED ?( ) ITS EXPLICIT EXPRESSION, WHICH WE WILL NEED LATER IN THE BOOK, IS ?(X) = (2?)?1/2 EXP ¡ ?1?2 X2¢ (1 06) UNLIKE ?( ), THE CDF, USUALLY DENOTED ?( ), HAS NO ELEMENTARY CLOSED-FORM EXPRESSION HOWEVER, BY (1 05) WITH A = ??AND B = X, WE HAVE ?(X) = Z X ?? ?(Y) DY THE FUNCTIONS ?( ) AND ?( ) ARE GRAPHED IN FIGURE 1 1 SINCE THE PDF IS THE DERIVATIVE OF THE CDF, IT ACHIEVES A MAXIMUM AT X = 0, WHERE THE CDF IS RISING MOST STEEPLY AS THE CDF APPROACHES BOTH 0 AND 1, AND CONSEQUENTLY, BECOMES VERY ?AT, THE PDF APPROACHES 0 ALTHOUGH IT MAY NOT BE OBVIOUS AT ONCE, DISCRETE RANDOM VARIABLES CAN BE CHARACTERIZED BY A CDF JUST AS WELL AS CONTINUOUS ONES CAN BE CONSIDER A BINARY R V X THAT CAN TAKE ON ONLY TWO VALUES, 0 AND 1, AND LET THE PROBABILITY THAT X = 0 BE P IT FOLLOWS THAT THE PROBABILITY THAT X = 1 IS 1?P THEN THE CDF OF X, ACCORDING TO THE DE?NITION OF F(X) AS PR(X ?X), IS THE FOLLOWING DISCONTINUOUS, “STAIRCASE” FUNCTION: F(X) = ( FOR X < P FOR 0 ?X < FOR X ?1 THIS CDF IS GRAPHED IN FIGURE 1 2 OBVIOUSLY, WE CANNOT GRAPH A CORRE- SPONDING PDF, FOR IT DOES NOT EXIST FOR GENERAL DISCRETE RANDOM VARIABLES, THE DISCONTINUITIES OF THE CDF OCCUR AT THE DISCRETE PERMITTED VALUES OF X, AND THE JUMP AT EACH DISCONTINUITY IS EQUAL TO THE PROBABILITY OF THE CORRESPONDING VALUE SINCE THE SUM OF THE JUMPS IS THEREFORE EQUAL TO 1, THE LIMITING VALUE OF F, TO THE RIGHT OF ALL PERMITTED VALUES, IS ALSO 1 REGRESSION MODELS