Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 201of no surprise. As the number of steps increases, it is expected that position of theparticle becomes normally distributed in the limit.7.2.2 PROBABILITY TABULATIONSOwing to its importance, we are often called upon to evaluate probabilitiesassociated with a normal random variable X:N(m, 2 ), such asZ " #1 b…x m† 2P…a < X b† ˆexp…2† 1=2 a 2 2 dx: …7:20†However, as we commented earlier, the integral given above cannot be evaluatedby analytical means and is generally performed numerically. For convenience,tables are provided that enable us to determine probabilities such as the oneexpressed by Equation (7.20).The tabulation of the PDF for the normal distribution with m ˆ 0 and ˆ 1is given in Appendix A, Table A.3. A random variable with distribution N(0, 1)is called a standardized normal random variable, and we shall denote it by U.Table A.3 gives F U (u) for points in the right half of the distribution only (i.e.for u 0). The corresponding values for u < 0 are obtained from the symmetryproperty of the standardized normal distribution [see Figure 7.6(a)] by therelationshipF U … u† ˆ1 F U …u†: …7:21†First, Table A.3 in conjunction with Equation (7.21) can be used to determineP(a < U b) for any a and b. Consider, for example, P 1:5 < U 2:5). It isgiven byP… 1:5 < U 2:5† ˆF U …2:5† F U … 1:5†:The value of F U (2: 5) is found from Table A.3 to be 0.9938; F U ( 1:5) is equal to1 F U 1:5), with F U 1:5) ˆ 0:9332, as seen from Table A.3. ThusP… 1:5 < U 2:5† ˆF U …2:5† ‰1 F U …1:5†Šˆ 0:994 1 ‡ 0:933 ˆ 0:927:More importantly, Table A.3 and Equation (7.21) are also sufficient fordetermining probabilities associated with normal random variables with arbitrarymeans and variances. To do this, let us first state Theorem 7.2.TLFeBOOK