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# W3-slides1 - Lecture 7- Outline and Examples Discrete vs...

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Lecture 7- Outline and Examples Discrete vs Continuous probability functions Random variables -deﬁnition -cumulative distribution function Discrete Random variables -deﬁnition -probability mass function -expected value -variance

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Discrete case ( Ross chapter 4) So far we deﬁned probability functions on discrete sample spaces (i.e. sample sets which are ﬁnite or countable). In a discrete sample space the (discrete) probability function assigns value P ( s ) for each outcome and we have: (a) P ( s ) 0 (b) all s S P ( s ) = 1. For any event A we have P ( A ) = all s A P ( s )
Continuous case (Ross chapter 5 Suppose the sample space S is an interval of real numbers.

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Continuous case (Ross chapter 5 Suppose the sample space S is an interval of real numbers. Example. Experiment: Measure lifetime of a light bulb. Here S = { x : x 0 } Example. Experiment: A student is picked at random from population of PSTAT 120A students and his/her weight is recorded. Here S = { x lbs : 100 x 250 } Here sample space has uncountably many outcomes.
When sample space has uncountably many outcomes, we cannot assign a positive value to every outcome (and have total probability 1). Here events are intervals of real numbers. We need a new way to assign probabilities to events. Easiest way- via integrals . We will use calculus to compute probabilities. Let f be a real-valued function deﬁned on S such that (a) f ( x ) 0 for all x S (b) R S f ( x ) dx = 1. Such a function is called probability density . Continuous probabilities are such that for any event A on S P ( A ) = Z A f ( x ) dx (thus, probability is deﬁned as an area under the graph of f .)

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Note : In continuous case f ( x ) is a probability density
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## This note was uploaded on 06/05/2010 for the course STAT PStat 120a taught by Professor Rohinikumar during the Spring '10 term at UCSB.

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W3-slides1 - Lecture 7- Outline and Examples Discrete vs...

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