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# Intro Prob & Stat for Business STT 315

MSU

GPA 3.72

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This 6 page Class Notes was uploaded by Aryanna Jerde on Saturday September 19, 2015. The Class Notes belongs to STT 315 at Michigan State University taught by Staff in Fall. Since its upload, it has received 41 views. For similar materials see /class/207819/stt-315-michigan-state-university in Statistics and Probability at Michigan State University.

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Date Created: 09/19/15

STT 315 Fall 06 Additional preparation for week of Wednesday September 6 2006 NOTICE Your point total for BOTH weeks one and two assignments combined will be 140 of the largest of the two scores plus 60 of the smaller of the two scores This is your chance to catch up if you did not do well on week one 1 Assume PA 08 PB 04 and A is independent of B a Give the complete Venn diagram for this information PAB 08 04 7032 PAB 08 06 048 PACB 702 04 008 PACBC 02 06 012 total ofwhich is 1 Sketch a loop labeled S sample space and within it two smaller loops which overlap labeled A and B respectively This de nes the four regions above Label these four regions with their appropriate names e g AB and their probabilities eg 032 b Give the complete tree diagram for this information FOR INDEPENDENT EVENTS A B CONDITIONAL PROBABILITIES DON T DEPEND UPON THE CONDITIONING EG PB l A PB SO THE SECOND TIER BRANCHES OF THE TREE DIAGRAM HAVE JUST PB OR PBC 04 B 032 AB 08 A 06 Bc 048 ABc drawin the lines of the tree 04 B 008 ACB 02 Ac 06 BC 012 ACBc total ofwhich is l 2 Assume PA 08 PB 04 and PBl A 03 a Give the complete Venn diagram for this information PAB 08 03 7024 PAB 7 PA 7 PAB 7 08 7 024 7 056 PACB 7 PB 7 PAB 7 04 7 024 7 016 PACBC l 7 024 7 056 7 016 004 total must equal 1 b Give the complete tree diagram for this information 03 B 08 A 07 Bc 01602 B 02 Ac 00402 Bc 024 056 016 004 PB l A 03 is given PBc l A 1 703 07 drawin the lines of the tree since PB i A PA BPA since PB i A PA B PA c From a or b determine PAl B THIS IS BAYES CALCULATION PAl B PABPB 024 04 06 differs from PA 08 3 Given the information that PAB 01 PAB 03 so PA 01 03 04 PACB 02 PACB 04 a Make a complete Venn diagram for this information JUST PLACE THE ABOVE IN THE FOUR REGIONS OF VENN S OVALS b Make a complete tree diagram for this information 14 B 04 A 34 Bc 13 B 06 Ac 23 Bc 01 03 02 04 PB l A PAB PA 0104 14 PBclA1714 34 drawin the lines of the tree PB i A PA BPA 02 06 13 PB i A PA B PA 1 7 13 23 4 A letter will be selected from the alphabet according to classical equal likely probability a Pletter is a vowel Pa e i 0 or u 526 b Pletter is a vowel l letter is after m in the alphabet Poorulafterm2l3 5 Given that PB l A 06 PA 07 and PB 02 Determine PAl B Proceeding formally PA l B PAB PB PA PB l A PB 07 0602 gt1 Therefore the given information conceals a contradiction is impossible 6 What is the underlying cause of the false positive paradox Even an accurate test ie one whose rates of false positives and false negatives are both low can be overwhelmed by applying it to a pool of subjects that are overwhelmingly negative eg free of disease 7 A number of experts settle on the following Pmedium term demand is rising 06 PMarch orders will be up l medium term demand is rising 08 PMarch orders will be up l medium term demand not rising 04 If we find that March orders rise what probability shall we then quote for a rise in medium term demand 8 Here is the probability distribution of X R 7 G where R and G refer to the numbers thrown by red and green dice It has been obtained by listing all 36 possibilities for the two dice and finding each possible value X of X and its probability pX The distribution is the list of all X and pX x pX RG 1 2 3 4 5 6 5 136 1 0 1 2 3 4 5 4 236 2 1 0 41 2 3 4 3 336 3 2 1 0 1 2 3 2 436 4 3 2 1 0 1 2 1 536 5 4 3 2 1 0 1 0 636 6 5 4 3 2 1 0 1 536 2 436 3 336 4 236 5 136 Plot the distribution where the horizontal aXis is X ie place a vertical bar of height pX at each ofthe values X 5 4 4 5 9 Refer to 8 a Calculate E R E G 35 ie E R 116 616 35 Con rm E X 0 using the additivity of expectation E EXER7GER7EG357350 b Determine E X from your distribution in 8 E X sum of X pX 5 136 5 136 0 c Determine E 2R 7 G using additivity of E E2R7G2ER7EG235735 d Determine ER 7 2 using additivity of E Think of 2 as a random variable that is constant ER72ERE23572 15 10 Given the distribution y py ypy yzpy yeEY2py 4 02 08 32 47720818 6 01 06 36 01 8 07 56 448 07 totals 10 70 516 26 a Determine E Y EYsumofypy7 b Determine Variance Y EY 7 E Y2 Variance on sum of y 7 E Y2 py 26 c Determine Variance Y E Y2 7 E Y2 which must agree with b Variance on 7 also E Y2 7 E Y2 7 516 7 72 7 26 d Determine the standard deviation SD of Y root of Variance Y from c or b SD on root26 11 A business venture has random return X with E X 12000 and Variance X 8100 a Determine E X 7 12000 EX 712000 E X 712000 12000 712000 0 b Determine standard deviation of X SD X root8100 90 c Determine the standard deviation of X 900 SDX900SDXl900190900110 12 Random variables X Y are INDEPENDENT with E X 2 and E Y 5 a Determine E X Y does not require independence EXYEXEY257 b Determine E X Y requires independence for INDEPENDENT rv E XY E X E Y 2 3 6 c If Variance X 41 and Variance Y 69 determine Variance X Y requires independence for INDEPENDENT rv VarianceX Y Variance X Variance Y 41 69 Sometimes this holds for nonindependent rv also in which case they are called UNCORRELATED But it always holds for independent rv 13 IQ has a bell distribution with mean 100 and standard deviation 15 Sketch this distribution and identify 68 and 95 intervals sketch a bell with mean 100 and sd 15 the interval from IQ 10015 to 100 15 is a 68 interval the interval from 100215 to 100 215 is a 95 interval the last two are approximations of the more accurate 95 interval 100 196 15 14 The Poisson distribution will be studied later in the course It often applies to the number X of occurrences of rare events an interesting feature of the Poisson distribution is that its standard deviation is the square root of its mean Furthermore although it is a discrete distribution the Poisson is rather well approximated by the normal provided the mean is around 3 or more The number X of auto accidents in a particular period is thought to be Poisson distributed with expectation 256 Sketch the approximate bell distribution for X and identify a 34 interval The Poisson law of rare events applies so the standard deviation root256 Sketch a bell with mean 256 and SD root256 The interval 256 to 256 root256 is an example ofa 34 interval interval mean mean sd is 34 of normal 15 The probability of OIL is 0 15 The probability of obtaining a positive test result if oil is present is 096 The probability ofa positive test result if oil is not present is 02 false positive a Express the above in proper notation POIL 015 P OIL 096 P l OIL 02 false negative b Give a complete tree diagram for the given information 096 POIL POIL P OIL 015 096 015 OIL 004 POIL POIL P OIL 015 004 drawin the lines of the tree 02 POILC POILC P OILC 085 02 085 01Lc 08 POILc POILC P OILC 085 08 c Determine P P POIL POIL total probability 015 096 085 02 d The test has shown positive What is our revised probability that there is oil given this information That is determine POIL l POIL l POIL P definition of conditional probability multiplication rule wtih P moved to the other side POIL POIL P 015 096 015 096 085 02 04586 an increase from POIL 015 due to the positive test result seen 16 Refer to 15 Suppose

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