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Calculating Variances of Various Random Variables Using PGF, MGF, and Characteristic Functions

This guide explores the calculation of variances for different types of random variables (RVs) including binomial, discrete uniform, Poisson, Gaussian, and Exponential RVs. It details the use of probability generating functions (PGFs) to find the variance of a binomial RV and a discrete uniform RV, as well as the moment generating function (MGF) for a Poisson RV. Additionally, it covers finding the characteristic function and variance of exponential and Gaussian RVs, including the standard N(0,1) case. Master these essential techniques in probability theory!

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Calculating Variances of Various Random Variables Using PGF, MGF, and Characteristic Functions

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  1. 4 transform methods

  2. pgf • 4-1: X is a binomial RV, B(n,p). Use pgf to calculate the variance of X. • 4-2: X is a (discrete) uniform RV that assumes integers between 1 and M. Calculate VAR[X] using its pgf.

  3. mgf • 4-3: Find the variance of a Poisson RV with parameter  using mgf.

  4. HW4-1 • Findthe mgf of a Gaussian RV X with N(0,1)

  5. Char. function • 4-4: Find the Variance of an Exponential RV X by using characteristic function. • 4-5: Find a characteristic function of a Gaussian RV, N(0,1).

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