# Topic 11 d the Normal Distribution
     #  find  P X < 1.45 )
pnorm( 1.45 )
     #  find P( X  < -0.28 )
pnorm( -0.28 )
     #  find P( X > -1.23 )  but that  = 1 - P( X < -1.23)
1 - pnorm( -1.23 )   # old way
pnorm( -1.23, lower.tail = FALSE)    # better way
     # find P( X > 0.66 ) but that is 1 - P(X < 0.66 )
1 - pnorm( 0.66 )   # old way
pnorm( 0.66, lower.tail=FALSE )   # better way
     # find P( X < 1.78 or X > 2.13 )
     # that is same as  P(X < 1.78 ) + P( X > 2.13 )
     # that is same as  P(X < 1.78 ) + (1 - P( X< 2.13 ) )
pnorm( 1.78 ) + (1 - pnorm( 2.13 ) )  # old way
pnorm( 1.78 ) + pnorm( 2.13, lower.tail=FALSE)  # better way
     # find P( -0.57 < X < 1.03 )
     # that is find P( X < 1.03 ) - P( X < -0.57 )
pnorm( 1.03 ) - pnorm( -0.57 )
#-------------------------------------------------------

     # find y such that P( X < y ) = 0.23
qnorm( 0.23 )
     # find y such that P( X > y ) = 0.075
qnorm( 1-0.075 )   # this is the ugly old way
qnorm( 0.075, lower.tail = FALSE )  # the better way
     # find y such that P( X < -y or X > y ) = 0.156
     # Because this is a symmetric distribution we can
     # just find y such that P( X > y ) = 0.156/2
qnorm( 0.56/2, lower.tail = FALSE)
     # find y such that P( -y < X < y ) = 0.925
     # By symmetry this is y such that P( X > y ) = (1-0.925)/2
qnorm( (1-0.925)/2, lower.tail = FALSE )
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