Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
EDEXCEL STATISTICS 1 PROBABILITY Discrete Uniform Distribution The Discrete uniform distribution is possibly the easiest of any discrete probability distribution to deal with. Common examples would be for instance Let X = the number uppermost on a fair unbiased dice Let X = the number that a fair unbiased 10 sided spinner lands on x=1,2,…….10 In each case the probability of each outcome is equally likely, with the PDF written as P( X x ) 1 n x 1,2,......n A fairground dartboard is divided into 9 equal sectors, a dart is thrown until it lands in the board, if this is modelled by a discrete uniform distribution then Let X= the number of the sector that the dart lands in x P(X=x) 1 1 9 2 1 9 3 1 9 4 1 9 5 1 9 6 1 9 7 1 9 8 1 9 9 1 9 (i) Write down E( X ) , Var( X ) (ii) Comment on the suitability of this model Ok, here’s a tip any question that says “write down” or “state” values should usually suggest that its either obvious, or there is a shortcut. In this case because of the symmetry of the distribution it is easily proved- and this is available in most textbooks eg. ref S1 p163/4 the following general result . For any discrete uniform distribution where P( X x ) Then E( X ) 1 n x 1,2,......n n 1 (n 1)(n 1) and Var( X ) 2 12 (i) 9 1 2 10 2 5 E( X ) Titus Salt School - A Teachnet Uk 2008 Project Page 1 of 2 EDEXCEL STATISTICS 1 PROBABILITY Discrete Uniform Distribution (9 1)(9 1) 12 80 12 20 3 Var ( X ) (ii) Whats wrong with this model ? – basically think of common sense reasons as to why the probability of hitting each number may not be equally likely. Assumes that the thrower simply aims randomly A real dart game would depend on the skill of the thrower etc Furthermore consider this extension to the question; The fair charges £1 per throw, and pays out (20 x no scored) - 10 (in pence) Find the average payout per throw Let P = payout in pence Then P=20X – 10 E(P ) E(20 X 10) 20E( X ) 10 20 5 10 90 In other words the fair charges £1 a throw and pays out £ 0.90 on average This makes use of the following general results for Expectation and Variance of a linear function of a random variable. E(aX b) aE( X ) b Var(aX b) a2Var( X ) Examples of applying these results can be found in the worked solutions to the Edexcel exam style questions. Titus Salt School - A Teachnet Uk 2008 Project Page 2 of 2