Generalize the minimax decision rule in order to classify patterns from three categories having triangle densities as follows: where δi > 0 is the half-width of the distribution (i = 1, 2, 3). Assume for convenience that µ1 <>2 <>3, and make some minor simplifying assumptions about the δi’s as needed, to answer the following: (a) In terms of the priors P(ωi), means and half-widths, find the optimal decision points x∗ 1 and x∗ 2 under a zero-one (categorization) loss. (b) Generalize the minimax decision rule to two decision points, x∗ 1 and x∗ 2 for such triangular distributions. (c) Let {µi, δi} = {0, 1}, {.5, .5}, and {1, 1}. Find the minimax decision rule (i.e., x∗ 1 and x∗ 2) for this case. (d) What is the minimax risk?

Use the standard error to construct a approximate prediction interval for Y using an alpha of 5%. (Round your answer to 2 decimal places   x x               Fuel Economy of Selected Vehicles (n = 73, k = 4)                           Obs Vehicle CityMPG Length Width Weight ManTran 1 Acura TL 20 109.3 74.0 3968 0 2 Audi A5 22 108.3 73.0 3583 1 3 BMW 4 Series 428i 22 182.6 71.9 3470 0 4 BMW X1 sDrive28i 23 176.5 70.8 3527 0 5 Buice LaCrosse 18 196.9 73.1 3990 0 6 Buick Enclave 17 201.9 79.0 4724 0 7 Buick Regal 21 190.2 73.1 3692 0 8 Cadillac ATS 22 182.8 71.1 3315 0 9 Cadillac CTS 20 195.5 72.2 3616 0 10 Cadillac Escalade 14 202.5 79.0 5527 0 11 Chevrolet Camaro 1SS 16 190.6 75.5 3719 1 12 Chevrolet Cruze LS 26 181.0 70.7 3097 0 13 Chevrolet Impala LTZ 19 201.3 73.0 3800 0 14 Chevrolet Malibu 2LT 25 191.5 73.0 3532 0 15 Chevrolet Spark LS 31 144.7 61.0 2269 1 16 Chevrolet Suburban LTZ 15 224.0 80.5 5674 0 17 Chrysler 200 Touring LX 20 191.7 72.5…

When using the central limit theorem for means with n = 57, it is not necessary to assume the distribution of the population data is normally distributed. True False

A 99 percent one-sample z-interval for a proportion will be created from the point estimate obtained from each of two random samples selected from the same population: sample R and sample S. Let R represent a random sample of size 1,000, and let S represent a random sample of size 4,000. If the point estimate obtained from R is equal to the point estimate obtained from S, which of the following must be true about the respective margins of error constructed from those samples? (A) The margin of error for S will be 4 times the margin of error for R. (B) The margin of error for S will be 2 times the margin of error for R. (C) The margin of error for S will be equal to the margin of error for R. (D) The margin of error for R will be 4 times the margin of error for S. (E) The margin of error for R will be 2 times the margin of error for S.

given normally distributed sample x=12 and s=3, use the Empirical Rule determine the upper and lower bounds to contain approximately 95% of your data

Use the standard error to construct a approximate prediction interval for Y using an alpha of 5%. (Round your answer to 3 decimal places.) Obs Assessed Floor Offices Entrances Age Freeway 1 1796 4790 4 2 8 0 2 1544 4720 3 2 12 0 3 2094 5940 4 2 2 0 4 1968 5720 4 2 34 1 5 1567 3660 3 2 38 1 6 1878 5000 4 2 31 1 7 949 2990 2 1 19 0 8 910 2610 2 1 48 0 9 1774 5650 4 2 42 0 10 1187 3570 2 1 4 1 11 1113 2930 3 2 15 1 12 671 1280 2 1 31 1 13 1678 4880 3 2 42 1 14 710 1620 1 2 35 1 15 678 1820 2 1 17 1 16 1585 4530 2 2 5 1 17 842 2570 2 1 13 0 18 1539 4690 2 2 45 0 19 433 1280 1 1 45 1 20 1268 4100 3 1 27 0 21 1251 3530 2 2 41 1 22 1094 3660 2 2 33 0 23 638 1110 1 2 50 1 24 999 2670 2 2 39 1 25 653 1100 1 1 20 1 26 1914 5810 4 3 17 0 27 772 2560 2 2 24 0 28 890 2340 3 1 5 0 29 1282 3690 2 2 15 1 30 1264 3580 3 2 27 0 31 1162 3610 2 1 8 1 32 1447 3960 3 2 17 0

To test for any significant difference in the number of hours between breakdowns for four machines, the following data were obtained. Machine1 Machine2 Machine3 Machine4 6.6 9.1 10.9 9.6 8.1 7.8 9.9 12.4 5.6 9.7 9.4 11.7 7.8 10.3 10.1 10.6 8.8 9.5 8.8 10.9 7.5 10.0 8.5 11.4 Use Fisher's LSD procedure to test for the equality of the means for machines 2 and 4. Use a 0.05 level of significance.   Find the value of LSD. (Round your answer to two decimal places.) LSD =   Find the pairwise absolute difference between sample means for machines 2 and 4.   x2 − x4= What conclusion can you draw after carrying out this test? There is a significant difference between the means for machines 2 and 4. There is not a significant difference between the means for machines 2 and 4.

If the random variable X follows the uniform distribution U= (0,1) What is the distribution of the random variable Y= -2lnX. Show its limits.

Suppose a shop owner claims that an equal number of customers come into his shop each weekday. To test this hypothesis, he records the number of customers that come into the shop on a given week and finds the following: Monday: 50 customers Tuesday: 60 customers Wednesday: 40 customers Thursday: 47 customers Friday: 53 customers He can use a Chi-Square Goodness of Fit Test to determine if the distribution of the customers that come in each day is consistent with his hypothesized distribution.

If X is a uniformly distributed random varibale with a=8 and b=13, then   Calculate the mean and variance of X?    Round to three decimal places