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**Extra info for Elements of Applied Probability for Engineering, Mathematics and Systems Science**

**Sample text**

Zn), ) X k E R X k where ,. over a variable means this variable i s omitted f r o m the sequence. Proof: We just prove the first statement. Note where 0 denotes the disjoint union. f. ’s given by p i . Later we shall see that the Xi’s are independent for the product probability P specified here. 3 Expectation We work up to a general definition of expectation by starting with simple random variables. e. R x is finite. 39 45 The expected value of a simple random variable X is This definition will be extended to general random variables but at this point let us just state that for discrete random variables this extension yields the same expression as above; that is z p x ( z ) provided EIXI EX = Izlpx(z) < 00.

S of X I ,X z , . . f. of the population p . This is the link between the real and perhaps unknown ( t o the statistician) distribution of the population and the sample which we observe. f. is that of the population. The sequence of random variables X I ,X z , . . d. sequence; that is the random variables are independent and identically distributed where we say random variables are independent if events generated by the individual Xi’s are independent in the sense given above. (Xi) be the collection of events in R of the form { X i E H i } where Hi is some subset of real numbers R.

9 The sample histogram the proportion of the sample which takes on the value z. 9. It is reasonable to guess that the sample proportions should be close to the population proportions. If we define the random variables e i to be 1 if X i = z and 0 otherwise, we have p e t ( l )= P(ei = 1) = p ( x ) and p,,(O) = P(ei = 0) = 1 - P ( Z ) . Moreover p n ( z ) = e i / n so E p n ( z )= (Eel . . Ee,)/n = Eel. However cy=l Eel = + + 1 . P ( X i = z) + 0 . P(X7,# z) = p ( z ) so p , ( z ) is an unbiased estimator of the population proportion.