CS723 - Probability and Stochastic Processes

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CS723 - Probability and Stochastic Processes

Lecture No. 17

In Previous Lectures Analysis of continuous random variables Uniform, exponential, Erlang, and Gaussian RV Gaussian random variable with PDF given by Sqrt(λ/π)e-λx2 has a variance of 1/2λ Gaussian PDF numerically integrated only for zero mean and unit variance RV Zero mean Gaussian RV with variance of 2 has a PDF Sqrt(1/2π2 ) e-(x//sqrt(2))2 Transform any Gaussian RV to equivalent zero mean unit variance Gaussian RV before numerical processing

Scaling of Gaussian RV

Standard Gaussian Events

Joint Distributions Observation of two continuous-valued quantities simultaneously PDF fXY(x,y) is a function of two variables (hill) representing relative likelihood of a certain pair of observed values There are marginal densities that can be obtained from the joint PDF Events are regions of XY plane and probability of an event is the volume of the hill over the region of the event The total volume of the hill must be equal to 1

Pair of Random Variables

Pair of Random Variables

Pair of Random Variables

Pair of Random Variables