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# Variable 0 1 2 where successes are relatively rare l

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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

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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, …)
Continuous random variables… l If assume these outcomes to be equally likely  each of the 60 outcomes has probability of 1/60 l Outcomes remain discrete l But if we can measure to any degree of accuracy  our rv now takes on any value in interval 0-60 l Have defined a continuous rv l For this rv need a different approach for assigning probabilities 22

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23 Continuous random variables… l Need to introduce concept of a probability density function (pdf) l This is a continuous version of a probability histogram used for discrete rv’s l Consider a continuous rv X with range a x ≤ b then its pdf f ( x ) must satisfy l f ( x ) ≥ 0 for all x between a & b l Total area under the curve between a & b is unity l Probabilities are now represented by areas under the pdf
24 Uniform random variable l The uniform pdf for our store delivery example would have the form l Graph of pdf would be l Equally likely nature of this rv is now represented by any interval of width m having equal probability = = otherwise 0 60 0 60 / 1 ) ( x x f x f (x) 60 0 1/60 m

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25 Uniform random variable… l The cumulative density function or cdf is defined as F ( x ) = P ( X x ) l For our store delivery example the cdf would have the form l Cdf is often called the distribution function of X < = 60 60 0 0 1 60 / 0 ) ( x x x x x F x F (x) 60 0 1
Uniform random variable… l Why was the height of the curve (constant) at 1/60? l What is the probability of any one (single) arrival time? 26

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Progress report #3 l Have now discussed random variables & probability distributions l Have introduced theoretical distributions that are useful in representing/modelling actual data l These cover both discrete distributions (binomial) & continuous (uniform) l Now ready to discuss the normal distribution l This distribution plays a pivotal role in statistics (both modelling and inference) l This is the classic bell-shaped distribution 27
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variable 0 1 2 where successes are relatively rare l Number...

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