Where Specifically,  = Probability P[ X < X ] = X = Random Variable P[ X < X1- ] = X = X1- = Random Variate P[ X > X1- ] = = Population Mean = E[X]  P[ X < ] = , thus,   =X0.5 2 = Population Variance = V[X] P[ X =  ] = and for any constant. = Population Standard Deviation  P[ X < X < X1- ] = Empirical Rule: P[ < X < 0.6826 - +] ≈ P[ < X < 0.9544 -2 +2] ≈ P[ < X < 0.9972 -3 +3] ≈ Standard Normal Probability Distribution Now consider the transformation, Z = ( X - ) / , where E[Z]=0 and V[Z]=1.   Thus, Z ~ N( 0 , 1 ), called the Standard Normal Distribution. Consider the graphs:   X X X  X~N( ,     Z  Z Z Z~N(  , Z~N( ,   Where Specifically,  = Probability P[ Z < Z ] = Z = Standard Normal Random Variable P[ Z < Z1- ] = Z = Z1- = Standard Normal Variate P[ Z > Z1- ] = = 0 = Population Mean  P[ Z < ] = , thus,   =X0.5 2 = 1 = Population P[ X =  ] = and for any constant. = 1 = Population Standard Deviation  P[ Z < Z < Z1- ] =
Where  = Probability X = Random Variable X = X1- = Random Variate  = Population Mean = E[X] 2 = Population Variance = V[X]  = Population Standard Deviation Specifically, P[ X < X ] = P[ X < X1- ] = P[ X > X1- ] = P[ X <  ] =  , thus, =X0.5 P[ X =  ] = and for any constant. P[ X < X < X1- ] = Empirical Rule: P[ - < X < +] ≈ 0.6826 P[ -2 < X < +2] ≈ 0.9544 P[ -3 < X < +3] ≈ 0.9972 Standard Normal Probability Distribution Now consider the transformation, Z = ( X -  ) /  , where E[Z]=0 and V[Z]=1. Thus, Z ~ N( 0 , 1 ), called the Standard Normal Distribution. Consider the graphs:   X  X X X~N(  ,    Z  Z Z Z~N(  , Z~N(  ,  Where  = Probability Z = Standard Normal Random Variable Z = Z1- = Standard Normal Variate  = 0 = Population Mean 2 = 1 = Population  = 1 = Population Standard Deviation Specifically, P[ Z < Z ] = P[ Z < Z1- ] = P[ Z > Z1- ] = P[ Z <  ] =  , thus, =X0.5 P[ X =  ] = and for any constant. P[ Z < Z < Z1- ] =
PM-2018-PERT-2.docx