Subject and UNIT: Random Process and Linear Algebra: Unit IV: Vector Spaces,,
Let V be a vector space and Let S a non-empty sub-set of V. A vector v ε V is called a linear combination of vectors of S if there exist a finite number of vectors
Subject and UNIT: Random Process and Linear Algebra: Unit IV: Vector Spaces,,
A subset W of a vector space V over a field F is called a sub-space of V if W is a vector space over F with the operations of addition and scalar multiplication defined on V.
Subject and UNIT: Random Process and Linear Algebra: Unit IV: Vector Spaces,,
The notation Mm x n or simply M, will be used to denote the set of all m x n matrices with entries from a field F. Then Mm x n is a vector space over F with respect to the usual operations of matrix addition and scalar multiplication of matrices.
Subject and UNIT: Random Process and Linear Algebra: Unit IV: Vector Spaces,,
In this section, we introduce the underlying structure of linear algebra, that of a finite dimensional vector space. The definition of vector space V, whose elements are called vectors, involves an arbitrary field F, whose elements are called scalars.
Subject and UNIT: Random Process and Linear Algebra: Unit III: Random Processes,,
Important Questions and Answers of Random Process
Subject and UNIT: Random Process and Linear Algebra: Unit III: Random Processes,,
Problems under n-step tpm pn
Problems of Markov chain
Subject and UNIT: Random Process and Linear Algebra: Unit III: Random Processes,,
Important problems of Markov Chain
Chapman Kolomogorov equations-Limiting distributions
Subject and UNIT: Random Process and Linear Algebra: Unit III: Random Processes,,
A discrete parameter Markov process is called a Markov Chain. example : The simple random walk moves, at each step, to a randomly chosen nearest neighbour.
Continuous time Markov Chain
Subject and UNIT: Random Process and Linear Algebra: Unit III: Random Processes,,
If X(t) represents the number of occurrences of a certain event in (0, t), then the discrete random process {X(t)} is called the Poisson process, provided that the following postulates are satisfied. (i) P[1 occurrence in (t, t+∆t)] = λ ∆t + 0 (∆t) (ii) P [0 occurrence in (t,t+t∆)] = 1 - λ ∆t + 0 (∆t) (iii) P [2 or more occurrences in (t, t+∆t)] = 0 (∆t) (iv) X(t) is independent of the number of occurrences of the event in any interval prior and after the interval (0, t). (v) The probability that the event occurs a specified number of times in (t0, t0+t) depends only on t, but not on to. Poisson process is not a stationary process, as its statistical properties (mean, autocorrelation, ...) are time dependent.
Subject and UNIT: Random Process and Linear Algebra: Unit III: Random Processes,,
"If the future value depends only on the present state but not on the past states is called a Markov process". X1, X2, ..., Xn-1, Xn are known as the states of the process, by our definition "future state Xn+1" depends only on "present state Xn", but not on "past states X1, X2, X3, ..., Xn-1
Subject and UNIT: Random Process and Linear Algebra: Unit III: Random Processes,,
Ergodic processes are processes for which time and ensemble (statistical) averages are interchangeable. The concept of ergodicity deals with the equality of time and statistical averages.
Jointly Wide-sense, N-th order Stationary, Example for WSS process,
Subject and UNIT: Random Process and Linear Algebra: Unit III: Random Processes,,
All SSS process is a WSS process but the converse is not true. i.e., Every WSS process need not be a SSS process.