A chi-squared random variable with ν > 0 degrees of freedom (χv2) has mgf M(t) = (1 − 2t) −ν/2 . Given that Z2 ∼ χ21, derive the mean and variance of Z2 using M(t). Confirm these results using the mgf of Z, namely MZ(t) = e1/2t2 .

A chi-squared random variable with ν > 0 degrees of freedom (χv2) has mgf M(t) = (1 − 2t) −ν/2 . Given that Z2 ∼ χ21, derive the mean and variance of Z2 using M(t). Confirm these results using the mgf of Z, namely MZ(t) = e1/2t2 .

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter1: Functions

Section1.2: The Least Square Line

Problem 5E

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Question

A chi-squared random variable with ν > 0 degrees of freedom (χ_v²) has mgf M(t) = (1 − 2t) −^ν/2 . Given that Z2 ∼ χ²₁, derive the mean and variance of Z² using M(t). Confirm these results using the mgf of Z, namely MZ(t) = e^1/2t².

A chi-squared random variable with v> 0 degrees of freedom (x²) has mgf M(t) =
(1 - 2t)-¹/2. Given that Z² ~ x², derive the mean and variance of Z² using M(t).
Confirm these results using the mgf of Z, namely Mz(t)
=
ezt²
=

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