This learning solution provides a clear exposition of essential tools of Random Variables, Two- Dimensional Random Variables, Random Processes, Queueing Models, Advanced Queueing Models, Correlation and Spectral Densities, Linear Systems with Random Inputs.
Probability - Introduction of probability - Set definitions - Operations on sets - Laws of algebra of sets. Experiments and sample spaces - Introduction of experiments and sample spaces. Discrete and Continuous sample spaces - Introduction of discrete and continuous sample spaces. Events - Introduction of events. Probability definitions and axioms - Introduction to probability Definitions and Axioms - Theorems on probability. Mathematical model of experiments - Mathematical (or) classical definition of probability - Example Problems - Mathematical tools - Example Problems. Probability as a relative frequency - Introduction to probability as a relative frequency. Joint Probability - Joint Probability - Example Problem. Conditional probability - Conditional probability - Introduction of conditional probability - Example Problems. Total probability and bay's theorem - Theorem of total probability - Baye's theorem - Example Problems. Independent Events - Independent events (multiplication law of probability) - Example Problems. Random variable - Introduction of random variable - Definition of random variable - Discrete, continuous and mixed random variable - Conditions for a function to be a random variable. Discrete Random Variable - Probability distribution function - Probability of mass function - Properties of discrete random variable - Example Problems. Continuous random variable - Probability distribution function - Probability density function - Properties of continuous random variable - Example Problems.
Binomial distribution - Introduction to binomial Distribution - Mean of binomial distribution - Variance of binomial distribution - Moment generating function - Example Problems. Poisson distribution - Introduction to poisson distribution - Mean of poisson distribution - Variance of poisson distribution - Moment of generating function - Properties of poisson distribution 1 - Properties of poisson distribution 2 - Example Problems. Uniform Distribution - Introduction to uniform distribution - Moment generating function of uniform distribution - Example Problems. Normal/Gaussian Distribution - Introduction to gaussian/Normal distribution - Application of gaussian distribution - Characteristics of the gaussian distribution - Normal distribution as a limiting form of binomial distribution - Example Problems. Exponential distribution - Introduction to exponential distribution - Exponential distribution - Example Problems - Memoryless property of the exponential distribution - Example Problems. Rayleigh density function - Introduction to rayleigh density function. Conditional distribution - Introduction to conditional distribution - Properties of conditional distribution. Conditioning event - Introduction to conditioning event. Conditional density function - Conditional density function - Conditional density properties - Example Problem. Introduction of a random variable - Introduction - Expected value of a random variable - Properties - Example Problems - Function of a Random Variable. Moments - Moments about the origin - Central moments. Variance and Skew - Introduction of variance - Properties of variance - Example Problems - Introduction of Skew. Chebychev’s inequality - Chebychev’s inequality - Example Problems. Characteristic function - Introduction of characteristic function - Properties of characteristic function - Example Problems. Moment generating function - Introduction of moment generating function - Property of moment generating function - Example Problems. Transformation of a random variable - Introduction - Monotonic transformation of a continuous random variable - Nonmonotic transformation of a continuous random variable - Transformation of a discrete random variable.
Random vector concept - Random vector concept - Sum of two random variables. Joint Distribution Function - Introduction - Properties - Example Problems. Joint Density Function - Introduction - Properties - Example Problems. Conditional Distribution and Density Function - Introduction of conditional distribution function - Properties of conditional distribution function - Introduction of conditioning event - Introduction of conditional density function - Properties of conditional density function - Point conditioning - Internal conditioning - Example Problems. Statistical independence of random variables - Introduction - Example Problems. Sum of random variables - Introduction - Two random variables - Multiple random variables - Example Problems. Central limit theorem - Introduction - Unequal distribution - Equal distribution. Expected value of a function of random variable - Expected value of a function of random variable - Example Problems. Joint moments - Joint moment about origin - Properties of correlation - Joint central moments - Properties of correlation - Example Problems. Joint characteristic function - Introduction of joint characteristic function - Properties of joint characteristic function - Theorem - Example Problems. Gaussian random variable - Two random variables - N random variables - Example Problem - Properties of gaussian random variables. Transformation of random variable - Transformation of random variable - Example Problem. Linear transformation of a gaussian random variable - Linear transformation of a gaussian random variable - Example Problems.
Least square curve fitting procedures - Introduction of Curve Fitting - Fitting a straight line - Non linear curve fitting - Example Problems. Covariance and Correlation - Covariance - Correlation - Correlation Analysis - Types of Correlation. Methods of Studying Correlation - Methods of Studying Correlation - Scatter diagram method - Graphic method - Karl Pearson's coefficient of correlation - Example Problems. Correlation of Grouped Data - Correlation of Grouped Data - Example Problems. Rank Correlation - Rank Correlation - Example Problems - Repeated Ranks - Example Problem. Regression Analysis - Regression Analysis - Example Problem. Angle Between Two Regression Lines - Angle Between Two Regression Lines - Example Problem. Transformation of 1D random variable - Transformation of 1D random variable - Working rules - Example Problem. Transformation of 2D random variable - Transformation of 2D random variable - Working rules - Example Problem.
Random Processes - Introduction - Classification of random process - Distribution and density functions - Statistical averages. Stationary random process - Stationary random process - Example Problem. Ergodic random process - Ergodic random process - Time averages and ergodicity - Mean ergodic process - Example Problem. Auto correlation function - Correlation techniques - Auto covariance - Properties Auto-correlation - Example Problem. Cross correlation function - Cross correlation - Properties of Cross Correlation - Example Problem. Linear systems - Introduction - Linear and time invariant systems - Impulse response of the linear systems - Linear time invariant systems - Transfer function of LTI system - Causal systems - Stable systems - Ideal systems - Example Problems. Linear system with random inputs - Introduction - System response - Mean value of output response - Mean square value of output response - Auto - correlation function of response - Cross - correlation function of input and output - Properties - Example Problems. Gaussian random process - Gaussian random process. Poisson random process - Introduction - Steady state and transient state systems - Poisson process - Arrival theorem - Example Problems. Power spectral density - Power Density Spectrum - Average Power of the Random Process - Auto correlation Function(ACF) of a Random Process - Example Problems - Properties of the Power Density Spectrum - Wiener Khinchine Theorem (Property 6) - Example Problem - Bandwidth of the Power Density Spectrum - Example Problems. Cross-power density spectrum - Cross-power density spectrum - Average cross power - Properties of cross power density spectrum - Example Problems - Relationship between cross power spectrum and cross correlation function - Example Problems. Spectral Characteristics of System Response - Introduction - Power Spectral Density of Y(t) - Cross Spectral Density of Input X(t) and Output Y(t) - Example Problems.
Queuing Theory - Introduction. Birth and Death Process - Pure Birth and Death Process - Derivation of Balance Equations. Model I (M/M/1): M/M/1 Queueing Model -Parameters of M/M/1 Model - Parameters of M/M/1 Model - Parameters of M/M/1 Model - Little Formulae and Facility Utilization - Example Problems. Model II (M/M/c): Multi server, infinite capacity queue - Average number of customers in the queue(lq) - Probability that ana arrival has to wait Average number of customer who have to actually wait (lw) - Example Problems. Model III (M/M/1): (N/FIFO) - Model III (M/M/1): (N/FIFO) - Model IV (M/M/C): (K/FCFS) - Model IV (M/M/C): (K/FCFS) - Example Problems. Finite source models - Non – Markovian Queueing Model - Pollaczek – Khinchine formula (PK - formula) - Characteristics for M/G/1 model - Example Problems. M/D/1 and M/EK/1 as special cases - Non exponential service: The M/D/1 Queue - Example Problem - Non exponential service: The M/EK/1 Queue - Example Problem. Networks - Networks - Open and Closed Networks. Series queues - Series queues (or) Tandem queues - Remark on two stage series queue - Example Problems. Series queues with blocking - Series queues with blocking - Example Problems. Open Jackson networks - Jackson networks - Open Jackson networks - Example Problems.
