Question 29.20: The continuous random variable x has a standard normal distr......

Engineering Mathematics A foundation for Electronic, Electrical, Communications and Systems Engineers [1040491]

The continuous random variable x has a standard normal distribution. Calculate the probability that

(a) x < 1.2 (b) x > 1.2 (c) x > −1.2 (d) x < −1.2

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(a) From Table 29.7
P(x < 1.2) = 0.8849
This is depicted in Figure 29.13.
(b) P(x > 1.2) = 1 − 0.8849 = 0.1151
This is shown in Figure 29.14.
(c) By symmetry P(x > −1.2) is identical to P(x < 1.2) (see Figure 29.15). So,
P(x > −1.2) = 0.8849
(d) Using part (c) we find
P(x < −1.2) = 1 − P(x > −1.2) = 0.1151
(see Figure 29.16).

Table 29.7 Cumulative normal probabilities.
z	A(z)	z	A(z)	z	A(z)	z	A(z)	z	A(z)	z	A(z)
0.00	0.500 000 0	0.40	0.655 421 7	0.80	0.788 144 6	1.20	0.884 930 3	1.60	0.945 200 7	2.00	0.977 249 9
0.01	0.503 989 4	0.41	0.659 097 0	0.81	0.791 029 9	1.21	0.886 860 6	1.61	0.946 301 1	2.01	0.977 784 4
0.02	0.507 978 3	0.42	0.662 757 3	0.82	0.793 891 9	1.22	0.888 767 6	1.62	0.947 383 9	2.02	0.978 308 3
0.03	0.511 966 5	0.43	0.666 402 2	0.83	0.796 730 6	1.23	0.890 651 4	1.63	0.948 449 3	2.03	0.978 821 7
0.04	0.515 953 4	0.44	0.670 031 4	0.84	0.799 545 8	1.24	0.892 512 3	1.64	0.949 497 4	2.04	0.979 324 8
0.05	0.519 938 8	0.45	0.673 644 8	0.85	0.802 337 5	1.25	0.894 350 2	1.65	0.950 528 5	2.05	0.979 817 8
0.06	0.523 922 2	0.46	0.677 241 9	0.86	0.805 105 5	1.26	0.896 165 3	1.66	0.951 542 8	2.06	0.980 300 7
0.07	0.527 903 2	0.47	0.680 822 5	0.87	0.807 849 8	1.27	0.897 957 7	1.67	0.952 540 3	2.07	0.980 773 8
0.08	0.531 881 4	0.48	0.684 386 3	0.88	0.810 570 3	1.28	0.899 727 4	1.68	0.953 521 3	2.08	0.981 237 2
0.09	0.535 856 4	0.49	0.687 933 1	0.89	0.813 267 1	1.29	0.901 474 7	1.69	0.954 486 0	2.09	0.981 691 1
0.10	0.539 827 8	0.50	0.691 462 5	0.90	0.815 939 9	1.30	0.903 199 5	1.70	0.955 434 5	2.10	0.982 135 6
0.11	0.543 795 3	0.51	0.694 974 3	0.91	0.818 588 7	1.31	0.904 902 1	1.71	0.956 367 1	2.11	0.982 570 8
0.12	0.547 758 4	0.52	0.698 468 2	0.92	0.821 213 6	1.32	0.906 582 5	1.72	0.957 283 8	2.12	0.982 997 0
0.13	0.551 716 8	0.53	0.701 944 0	0.93	0.823 814 5	1.33	0.908 240 9	1.73	0.958 184 9	2.13	0.983 414 2
0.14	0.555 670 0	0.54	0.705 401 5	0.94	0.826 391 2	1.34	0.909 877 3	1.74	0.959 070 5	2.14	0.983 822 6
0.15	0.559 617 7	0.55	0.708 840 3	0.95	0.828 943 9	1.35	0.911 492 0	1.75	0.959 940 8	2.15	0.984 222 4
0.16	0.563 559 5	0.56	0.712 260 3	0.96	0.831 472 4	1.36	0.913 085 0	1.76	0.960 796 1	2.16	0.984 613 7
0.17	0.567 494 9	0.57	0.715 661 2	0.97	0.833 976 8	1.37	0.914 656 5	1.77	0.961 636 4	2.17	0.984 996 6
0.18	0.571 423 7	0.58	0.719 042 7	0.98	0.836 456 9	1.38	0.916 206 7	1.78	0.962 462 0	2.18	0.985 371 3
0.19	0.575 345 4	0.59	0.722 404 7	0.99	0.838 912 9	1.39	0.917 735 6	1.79	0.963 273 0	2.19	0.985 737 9
0.2	0.579 259 7	0.60	0.725 746 9	1.00	0.841 344 7	1.40	0.919 243 3	1.80	0.964 069 7	2.20	0.986 096 6
0.21	0.583 166 2	0.61	0.729 069 1	1.01	0.843 752 4	1.41	0.920 730 2	1.81	0.964 852 1	2.21	0.986 447 4
0.22	0.587 060 4	0.62	0.732 371 1	1.02	0.846 135 8	1.42	0.922 196 2	1.82	0.965 620 5	2.22	0.986 790 6
0.23	0.590 954 1	0.63	0.735 652 7	1.03	0.848 495 0	1.43	0.923 641 5	1.83	0.966 375 0	2.23	0.987 126 3
0.24	0.594 834 9	0.64	0.738 913 7	1.04	0.850 830 0	1.44	0.925 066 3	1.84	0.967 115 9	2.24	0.987 454 5
0.25	0.598 706 3	0.65	0.742 153 9	1.05	0.853 140 9	1.45	0.926 470 7	1.85	0.967 843 2	2.25	0.987 775 5
0.26	0.602 568 1	0.66	0.745 373 1	1.06	0.855 427 7	1.46	0.927 855 0	1.86	0.968 557 2	2.26	0.988 089 4
0.27	0.606 419 9	0.67	0.748 571 1	1.07	0.857 690 3	1.47	0.929 219 1	1.87	0.969 258 1	2.27	0.988 396 2
0.28	0.610 261 2	0.68	0.751 747 8	1.08	0.859 928 9	1.48	0.930 563 4	1.88	0.969 946 0	2.28	0.988 696 2
0.29	0.614 091 9	0.69	0.754 902 9	1.09	0.862 143 4	1.49	0.931 887 9	1.89	0.970 621 0	2.29	0.988 989 3
0.30	0.617 911 4	0.70	0.758 036 3	1.10	0.864 333 9	1.50	0.933 192 8	1.90	0.971 283 4	2.30	0.989 275 9
0.31	0.621 719 5	0.71	0.761 147 9	1.11	0.866 500 5	1.51	0.934 478 3	1.91	0.971 933 4	2.31	0.989 555 9
0.32	0.625 515 8	0.72	0.764 237 5	1.12	0.868 643 1	1.52	0.935 744 5	1.92	0.972 571 1	2.32	0.989 829 6
0.33	0.629 300 0	0.73	0.767 304 9	1.13	0.870 761 9	1.53	0.936 991 6	1.93	0.973 196 6	2.33	0.990 096 9
0.34	0.633 071 7	0.74	0.770 350 0	1.14	0.872 856 8	1.54	0.938 219 8	1.94	0.973 810 2	2.34	0.990 358 1
0.35	0.636 830 7	0.75	0.773 372 6	1.15	0.874 928 1	1.55	0.939 429 2	1.95	0.974 411 9	2.35	0.990 613 3
0.36	0.640 576 4	0.76	0.776 372 7	1.16	0.876 975 6	1.56	0.940 620 1	1.96	0.975 002 1	2.36	0.990 862 5
0.37	0.644 308 8	0.77	0.779 350 1	1.17	0.878 999 5	1.57	0.941 792 4	1.97	0.975 580 8	2.37	0.991 106 0
0.38	0.648 027 3	0.78	0.782 304 6	1.18	0.880 999 9	1.58	0.942 946 6	1.98	0.976 148 2	2.38	0.991 343 7
0.39	0.651 731 7	0.79	0.785 236 1	1.19	0.882 976 8	1.59	0.944 082 6	1.99	0.976 704 5	2.39	0.991 575 8
z	A(z)	z	A(z)	z	A(z)	z	A(z)	z	A(z)	z	A(z)
2.40	0.991 802 5	2.45	0.992 857 2	2.50	0.993 790 3	2.55	0.994 613 9	2.60	0.995 338 3	3.20	0.999 312 9
2.41	0.992 023 7	2.46	0.993 053 1	2.51	0.993 963 4	2.56	0.994 766 4	2.70	0.996 533 0	3.40	0.999 663 1
2.42	0.992 239 7	2.47	0.993 244 3	2.52	0.994 132 3	2.57	0.994 915 1	2.80	0.997 444 9	3.60	0.999 840 9
2.43	0.992 450 6	2.48	0.993 430 9	2.53	0.994 296 6	2.58	0.995 060 0	2.90	0.998 134 2	3.80	0.999 927 7
2.44	0.992 656 4	2.49	0.993 612 8	2.54	0.994 457 4	2.59	0.995 201 2	3.00	0.998 650 1	4.00	0.999 968 3

										4.50	0.999 996 6
										5.00	0.999 999 7
										5.50	0.999 999 9