For example, if Xn represents the outcome of the nth toss of Now let S n= X 1 +X 2 +¢¢¢+X nbe the sum of nindependent random variables of an independent trials process with common distribution function mdeﬂned on the integers. point process. Poisson random variables is also Poisson. In electronics, white noise is defined as having a flat frequency spectrum ('white') and being random ('noise'). The textbook for this subject is Bertsekas, Dimitri, and John Tsitsiklis. X(t,w) is called a random process. Here again, knowing that the result is Poisson allows one to determine the parameters in the sum density. Recall that a Poisson density is completely speciﬁed by one number, the mean, and the mean of the sum is the sum of the means. If w is ﬁxed, X(t,w) is a deterministic time function, and is called a realization, a sample path, or a The sum of two S.I. A stochastic process corresponding to a sequence of random variables $\{ t _ {i} \}$, $\dots < t _ {-} 1 < t _ {0} \leq 0 < t _ {1} < t _ {2} < \dots$, on the real line $\mathbf R ^ {1}$. 2 It is easy to see that the convolution operation is commutative, and it is straight-forward to show that it is also associative. Classiﬁcation of Random Processes Depending on the continuous or discrete nature of the state space S and parameter set T, a random process can be classiﬁed into four types: 1. Deﬁnition of a Random Process Assume the we have a random experiment with outcomes w belonging to the sample set S.To each w ∈ S, we assign a time function X(t,w), t ∈ I, where I is a time index set: discrete or continuous. If both T and S are discrete, the random process is called a discrete random sequence. Our interest centers on the sum of the random variables for all the arrivals up to a fixed time $$t$$, which thus is a Poisson-distributed random sum of random variables. These variables are independent and identically distributed, and are independent of the underlying Poisson process. Noise generally can be contrasted with 'interference', one or more undesired signals being picked up from elsewhere and being added to the signal of interest, and 'distortion', undesired signals being generated from nonlinear processes acting on the signal of interest itself. Each value $t _ {i}$ corresponds to a random variable $\Phi \{ t _ {i} \} = 1 , 2 \dots$ called its multiplicity. of the random variable Z= X+ Y. • For ﬁxed ζ, the graph of the function X(t,ζ) versus t is a sample function of the random process. Introduction to Probability. Random Processes: Mean and Variance ... • The expected aluev of the sum of two or more random ariables,v is the sum of each individual expected alue.v E[X +Y] = E[X]+E[Y] (6) 2 Mean-Square alueV If we look at the second moment of the term (we now look at x2 in the integral), then Part III: Random Processes Download Resource Materials; The videos in Part III provide an introduction to both classical statistical methods and to random processes (Poisson processes and Markov chains). Y. S. Han Random Processes 1 Deﬁnition of a Random Process • Random experiment with sample space S. • To every outcome ζ ∈ S, we assign a function of time according to some rule: X(t,ζ) t ∈ I.