By Edward P. C.(Edward P.C. Kao) Kao
Meant for a calculus-based path in stochastic methods on the graduate or complex undergraduate point, this article deals a contemporary, utilized viewpoint. rather than the traditional formal and mathematically rigorous strategy ordinary for texts for this direction, Edward Kao emphasizes the improvement of operational abilities and research via quite a few well-chosen examples.
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Extra resources for An Introduction to Stochastic Processes
Independence 23 probability 1/2 of being present, each possible outcome has the same probability |Ω|−1 . We denote this probability measure by Pn . The event that the resulting network has a connection from left to right is denoted by LR(n). 12. It is the case that Pn (LR(n)) = 1 , 2 for all n. Proof. Since every outcome has the same probability, it is enough to show that the number of outcomes in which there is a connection from left to right, is the same as the number of outcomes for which there is no such connection.
9. However, there is a better way to compute P (Ak ). Note that we have constructed the experiment in such a way that the events Bi are independent. Indeed, we built our probability measure in such a way that any outcome with k 1s and n − k 0s has probability pk (1 − p)n−k , which is the product of the individual probabilities. Hence we see that P (Ak ) = = P (B1 ∩ B2 ∩ · · · ∩ Bk−1 ∩ Bkc ) P (B1 )P (B2 ) · · · P (Bk−1 )P (Bkc ) = (1 − p)k−1 p. 11 (Random networks). The theory that we discussed so far can sometimes be used in a very surprising way.
First observe that ⎛ ⎞ Pn (∪nk=0 Bk,n ) Pn ⎝ Bk,n ⎠ k>n( 12 + = Pn (Bk,n ) k>n( 12 + ) ) = k>n( 12 + ) n −n 2 . k 28 Chapter 1. Experiments The following surprising trick is quite standard in probability theory. 9. 2. Using this inequality, we ﬁnd that the last expression is at most 2 e−λn eλ /4 n 2 = eλ n/4−λn . Now we can ﬁnd λ to minimise the right hand side, that is, λ = 2 . 5) k>n( 12 + ) which tends to zero when n tends to inﬁnity. 2. Prove that for all x ∈ R, we have ex ≤ x + ex . 1. In this exercise, A and B are events.
An Introduction to Stochastic Processes by Edward P. C.(Edward P.C. Kao) Kao