Course Notes - Department of Mathematics and Statistics

5 Probability DistributionsFred’s Rugby TeamFred is a avid rugby player, and has started coaching using some innovativetechniques to improve pace. Out of Fred’s team of 15, 5 havedeveloped leg injuries. Fred wondered if maybe his coaching had somethingto do with this, so he went onto Google, and found a studythat suggested that 15% of rugby players develop leg injuries. Is thereenough evidence to suggest Fred’s coaching techniques are causing moreinjuries than normal?Fred also wanted to know if his new training techinique were working.He had read that 40 metre sprint times for players are normallydistributed with a mean of 7 seconds and a standard deviation of 0.6seconds. What times would his players need to record to be in thetop 10% of players? What is the probability a player could run the 40metres in 6.1 seconds?5.1 Binomial Distribution• Arises when investigating proportions e.g.population with diabetes.proportion of adult• Each individual has or does not have diabetes (binary outcome).• Let Y be the random variable for an individual outcome of aperson in the population.• There are two outcomes: Y = 1 and Y = 0 - generally speakingthis refers to Success and Failure respectively.• The parameter π represents the unknown proportion of 1’s occuring.Probability DistributionThe probability distribution of Y is:Y = y i Pr(Y=y i )1 (Success) π0 (Failure) 1 − π60