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00 000 004 013 025 039 017 002 note that we have 7 p

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Unformatted text preview: 0.25 0.39 0.17 0.02 Note that we have 7 P (X = x) = 1 x=0 ORF 245 Prof. Rigollet Fall 2012 Probability distribution This is true in general. From the addition rule for disjoint events, we have P (X = x) = P(X takes one of its possible values)=1 all possible x The probability distribution is a function p(x) defined by p(x) = P (X = x) and is sometimes called “probability mass function”. This function can also be described by a formula instead of enumerating all possible values x. Example: Flip a coin 1000 times. X=number p(x) = of heads. Possible values {0,...,1000}. 1 21000 1000 x ⇥ ORF 245 Prof. Rigollet Fall 2012 Expected Value For Probability, we had: As the number of observations goes to infinity, the proportion of occurrences of a given outcome converges to the probability of this outcome. We now have As the number of observations goes to infinity, the average of the observed values converges to the expected value of the random variable. 1 x= ¯ n n xi i=1 E (X ) ORF 245 Prof. Rigollet Fall 2012 Expected Value But we can rearrange the sum to get n 1X 1 xi = n i=1 n X x possible = X x · (number of xi equal to x) x possible x · (proportion of xi equal to x) xP (X = x) from the LLN x possible 1 ¯ Thus x = n n xi i=1 E (X ) = x possible xP (X = x) ORF 245 Prof. Rigollet Fall 2012 Expected Value The expected value of the discrete random variable X is defined by xP (X = x) E (X ) = x possible ORF 245 Prof. Rigollet Fall 2012 Expected Value The expected value of the discrete random variable X is defined by xP xP (X = x) E (X ) = x possible In the same way, we can define the expected value of the function h(X) of a discrete random variable X by ORF 245 Prof. Rigollet Fall 2012 Expected Value The expected value of the discrete random variable X is defined by xP (X = x) E (X ) = x possible In the same way, we can define the expected value of the function h(X) of a discrete random variable X by E ( h(X)) = x possible h( x ) P (X = x) ORF 245 Prof. Rigollet Fall 2012 Variance An interesting choice of the function h(X) is h(X ) = (X µ)2 where µ = E (X ) With this choice, we obtain the variance of X var(X ) = (x µ)2 P (X = x) x possible It is the limit of s2 as the number of observations goes to infinity. The following shortcut formula is useful: var(X ) = E (X 2 ) Standard Deviation v ar(X ) µ2 ORF 245 Prof. Rigollet Fall 2012 Example Going back to the number of courses example: x 2 3 4 5 6 7 P(X=x) 0.04 0.13 0.25 0.39 0.17 0.02 E (X...
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This document was uploaded on 10/14/2013.

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