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DISTRIBUTION FUNCTION OF THE NORMAL DISTRIBUTION The function tabulated is +() = e- du 0.02 0.03 0.01 0.07 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 05199 0.5239 0.5279 0.5019 0.5359 0.1 0.5398 0.5438 0.5478 0.5617 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.58.12 0.5571 0.5910 0.5945 0.5947 0.6026 0.6064 0.6103 0.6141 03 0.6179 0.6217 0.6255 0.6293 0.6331 0.6365 0.6-406 0,6443 0.6480 0.6517 04 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.680S 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7048 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7369 0.7422 0.7454 0.7486 0.7517 0.75-49 0.7 0.7580 0.7611 0.76-42 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7861 0.7910 0.7939 0,7967 0.7995 0.5023 0.8051 0,8078 0.8106 0.8133 0.9 D.8159 0.8186 0.8212 0.8238 0.6264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1. 0.8643 0.8665 0.8686 0.8708 0.6729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.6849 0.6869 0.8858 0.8907 0.8925 0.8944 0.8962 0.8960 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94-41 16 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 17 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.H 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9649 0.9706 0.9713 0.9719 0.9726 0.9732 0.9735 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.97725 0.97778 0.97831 0.97882 0.97932 0.97962 0.98030 0.98077 0.98124 0.98169 2.1 0.98214 0.98257 0.98300 0.98341 0.98382 0.98422 0.98-461 0,98500 0.98537 0.98574 2.2 0.98610 0.98645 0.98679 0.98713 0.98745 0.96778 0.98809 0.98840 0.93870 0.98899 2.3 0.98928 0.98956 0.98983 0.99010 0.99036 0.99061 0.99086 0,99111 0.99134 0.99158 2.4 0.99180 0.99202 0.99224 0.99245 0.99266 0.99286 0.99305 0.99324 0.993-43 0.99361 2.5 0.99379 0.99396 0.90413 0.99430 0.994 46 0.99461 0.99477 0,99492 0.99506 0.99520 2.6 0.99534 0.99547 0.99560 0,99573 0.99585 0.99598 0.99609 0,99621 0.99632 0.99643 17 0.99654 0.99664 0.99674 0,99683 0.99693 0.99702 0.99711 0,99720 0.99728 0.99736 2.8 0.99744 0.99752 0.99760 0.99767 0.99774 199761 0.99788 0.99795 0.99801 0.99807 20 0.99813 0.99819 0.9982 0.99831 0.99836 0.99841 0.99846 0.99851 0.99856 0.99861 3.01 0.99865 0.99869 0.99874 0,99878 0.99682 0.99856 0,99:889 0,99693 0.99896 0.99900 3.1 0.99903 0.90906 0.94910 0,99913 0.99916 0.99918 0.99921 0,99924 0.99926 0.99929 3.1 0.99931 0.99934 0.99936 0.99938 0,999 40 0.999.12 0.99944 0,999 46 0.999-48 0.99950 3.1 0.99952 0.90953 0.99957 0.99958 0.99960 0.99961 0.99962 0.99964 0.99915 3.4 0.99066 0.99969 0.99970 0.99971 0.99972 0.99973 0,99974 0.99975 0.99976 3.5 0.999767 0.999776 0.999784 0.999792 0.999500 0.999817 0.999815 0,999822 0.999828 0.999835 0.999841 0.999847 0.999853 0.999858 0.999564 0.999569 0.999874 0,999879 0.999683 0.999838 1.7 0.999692 0.999896 0.999900 0.999904 0.999908 0.999912 0.999915 0,999918 0.999922 0.999925 0.999925 0.999931 0.999933 0.999936 0.999938 0.999941 0.999943 0,999946 0.999948 0.999950 0,999953 0.999954 0.939956 0.999958 0.999959 0.999961 0.999963 0,999964 0.999966 0.999967The monthly rainfall in Chicago is (in inches) a random variable X which is normally distributed with a mean 1 and standard deviation 0.5. (i) What is the probability that Chicago has more than 1.75 inches rainfall in January? (ii) Find the probability density function of (X 1)2 (iii) Calculate the expectation of (X 1)2 . Hint: use the variance of X