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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.
 Fall '09

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