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lecture+4 - Random Variables and Expecta4on Random...

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Random’Variables’and’ Expecta4on’
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Random’Variable’(one’way’to’think’ about’them).’ Numeric’(?)’outcome’of’(random!)’observa4on.’ The’random’variable’induces’a’probability’setup’ The’sample’space’is’all’possible’outcomes’ P(A)’=’P(X’in’A)’
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Random’variable’ versus’ realiza’on ’of’a’random’variable’ Draw’a’student’from’the’class’at’random,’ record’their’height’in’inches.’The’height’is’a’ random’variable.’ If’we’actually’did’this,’the’recorded’height’ would’be’a’number.’That’number’would’be’ the’ realiza’on ’of’the’random’variable.’
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Discrete’versus’Con4nuous’ Discrete:’fnite’(or’ countably’infnite)’set’oF’ possible’outcomes.’ Con4nuous:’the’random’variable’takes’values’ in’a’con4nuum.’(Usually,’this’is’a’theore4cal’ approxima4on.)’
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Probability’Mass’Func4on’ (for’discrete’variable’only!)’ p(x)’=’P{X’=’x}’ F(x)’=’P{X’less’than’or’equal’to’x}’ Sum’of’p(x±)’for’x±’up’to’x.’ If’p(x)’sums’to’1,’and’all’elements’are’posi4ve,’ then’it’is’the’probability’mass’func4on’of’a’ random’variable…’
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Example’ Random’variable’ Draw’height’from’front’row’(now)’ Sample’space’ Depends’on’the’front’row’ Realiza4on’ We’have’to’pick’one’ The’probability’mass’func4on’ Equal’mass’at’each’point’ The’distribu4on’func4on’ Let±s’plot’it’
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Density’ (for’con4nuous’random’variables’only!)’ The’integral’of’f(x)’from’a’to’b’is’the’ probability’that’X’is’in’the’interval’[ a,b].’ Or’( a,b)’or’( a,b]’or’[ a,b).’ P(X=x)’=’0’for’con4nuous’random’variables’
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Example’ Flick’a’spinner,’and’record’the’angle’(from’0’ to’360’degrees)’of’the’exact’center’of’the’end’ of’the’pointer.’ What’would’be’the’density’func4on?’ What’would’be’the’distribu4on’func4on’
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