# Look like for n 10 p 02 l if with study chances of

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look like for n = 10, p = 0.2? l If with study, chances of correctly answering improves to p = 0.5? l What is the binomial distribution in this case? l What are the key features of these 2 distributions? l What would you expect for p = 0.8? Binomial distribution with n = 10, p = 0.2 0.00 0.10 0.20 0.30 0.40 0 1 2 3 4 5 6 7 8 9 10 Correct answers Probability Binomial distribution with n = 10, p = 0.5 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 1 2 3 4 5 6 7 8 9 10 Correct answers Probability

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16 Mean & variance of a binomial random variable = = = = - = = = = = = - = - + - = - + - - = = × + - × = - = n i i n i i n i i n i i i i i i i i p np X Var X Var X Var np X E X E X E p p p p p p p p p p X Var p p p X E p p x X P x X 1 1 1 1 2 2 ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( Thus ) 1 ( ) 1 )( 1 ( ) 1 ( ) 1 ( ) 0 ( ) ( 1 ) 1 ( 0 ) ( 1 1 0 ) ( by given is of on distributi The
Mean & variance of a binomial random variable… l In week 4 we calculated mean & variance of X in gender composition eg by first principles l Now have alternative method of calculation l E( X )= np =3×0.5=1.5 l Var( X )= np( 1 -p) =3×0.5×0.5=0.75 17 x 0 1 2 3 P ( X = x ) 0.125 0.375 0.375 0.125

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18 Models as approximations All models are wrong but some are useful George Box l In gender composition example l Define X = number of boys in families with 3 children l Assume l Birth outcomes are independent events l Probability does not vary over births l Males & females are equally likely to be born l First 2 assumptions required for binomial to be appropriate for X ? l How reasonable are all three assumptions?
19 Models as approximations… l Is the number of Swans wins in a football season likely to be well-approximated by a Binomial distribution & if not why not?

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20 Other common distributions l There are many other common distributions that can be used to represent or model real phenomena l The Poisson is another discrete probability distribution l Binomial – number of successes in sequence of trials l Poisson – number of successes in a period of time or region of space l Refers to a count variable (0, 1, 2, …) where successes are relatively rare l Number of times an individual visited a GP in last year l We now want to consider continuous rather than discrete rv’s
21 Continuous random variables l For discrete rv’s we assigned probabilities to different outcomes l In binomial model X = 0, 1, 2, …, n ( n finite) l In Poisson model X = 0, 1, 2, ... (said to be countably infinite) but associated rv remains discrete l Consider store deliveries that are confined to 7- 8am l Let rv of interest be number of minutes after 7 that deliveries are made l Thus outcomes would be 1, 2, …, 60 (7.01, 7.02, …)

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Continuous random variables… l If assume these outcomes to be equally likely  each of the 60 outcomes has probability of 1/60 l
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• Three '11
• DenzilGFiebig
• Probability, Probability distribution, Probability theory, probability density function, Cumulative distribution function

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