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The Islamic University of Gaza Faculty of Commerce Economics & Applied Statistics Department Mathematical stat. – Instructor : Ibrahem Abed Nov. 12th, 2009 Time: 1 hour Mid. Exam Question #1:( 5 points) let Y be a random variable with probability density function given by : 2 (1 y) f ( y) o , , 0 y 1 elsewhere By using the method of transformation , Find the probability density function of U 2Y 1 Question #2:( 10 points) a) If X denote a random sample with 60 and S 2 64 16 Find P( xi 992) i 1 b) Suppose that 2 1.4 . Let S 2 denote the sample variance of the 20 observations Find a number (a) such that P(a S 2 ) 0.975 Question #3:( 10 points) a) How many observations should be included in the sample such that the sample mean will be within 1 of the population mean , with probability 0.90 knowing that 2 16 ? b) Let S12 denote the sample variance for a random sample of ten values and S 22 denote the sample variance for the sample of eight values and S12 to be independent of S 22 such that 1 2 2 2 Find a number b such that P( S12 b) 0.95 S 22 1 Question #4 :( 10 points) Let Y1 ,Y2 be any two variables such that the joint density function of Y1 and Y2 given by 3 y f ( y1 ; y 2 ) 1 o , 0 y 2 y1 1 , elsewhere a) By using the method of Distribution Functions , Find the probability density function for U Y1 Y2 b) Find E (U ) Question #5:( 5 points) Let Y1 be a binomial random variable with n1 trails and probability of success given by p , Let Y2 be a binomial random variable with n 2 trails and probability of success given by p . and Y1 ,Y2 are independent . By using the method of moment generating functions,Find the density function for U Y1 Y2 02.025,20 02.025,19 02.025,18 02.05,18 34.1696 32.8523 31.5264 28.8693 P(Z > 2) Z 0.05 Z 0.025 Z 0.10 0.0228 1.645 1.96 1.28 Z 0.02 F0.05,9 , 7 F0.05, 7 ,9 F0.05,10,8 F0.05,8,10 F0.10, 7 ,9 F0.10,9 , 7 P(Z > 0) 2.05 3.68 3.29 3.35 3.07 2.51 2.72 0.5 P(Z > 1) df = 15 P(t <1) 0.8334 df = 16 P(t < 1) 0.8339 df = 14 P(t < 1) 0.8328 0.1587 02.975,20 02.975,19 02.975,18 02.95,18 9.590 8.906 8.230 9.39 Good luck ☺☺☺ 2