sta257week4notes - Relation between Binomial and Poisson...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
week 4 1 Relation between Binomial and Poisson Distributions Binomial distribution Model for number of success in n trails where P (success in any one trail) = p . Poisson distribution is used to model rare occurrences that occur on average at rate λ per time interval. Can think of “rare” occurrence in terms of p Æ 0 and n Æ . Take these limits so that λ = np . So we have that
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
week 4 2 Continuous Probability Spaces is not countable. Outcomes can be any real number or part of an interval of R , e.g. heights, weights and lifetimes. Can not assign probabilities to each outcome and add them for events. Define as an interval that is a subset of R . F – the event space elements are formed by taking a (countable) number of intersections, unions and complements of sub-intervals of . Example: = [0,1] and F = { A = [0,1/2), B = [1/2, 1], Φ , }
Background image of page 2
week 4 3 How to define P ? Idea - P should be weighted by the length of the intervals. - must have P ( ) = 1 - assign 0 probability to intervals not of interest. •F o r the real line, define P by a (cumulative) distribution function as follows: F ( x ) = P ((- , x ]). Distribution functions (cdf) are usually discussed in terms of random variables.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
week 4 4 Recalls
Background image of page 4
week 4 5 Cdf for Continuous Probability Space For continuous probability space, the probability of any unique outcome is 0. Because, P ({ ω }) = P (( ω , ω ]) = F ( ω ) - F ( ω ) = 0.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 22

sta257week4notes - Relation between Binomial and Poisson...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online