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7. Let X₁, X2,..., Xn be independent variables, X; being N(µi, 1), and let y = = x² + x² +...+X²/2. Show that the characteristic function of Y is by(t) = 1 (1-2it)n/2 exp ite 1-2it where 0 = ²+²+ ...+. The random variables Y is said to have the non-central chi-squared distribution with n degrees of freedom and non-centrality parameter 0, written x² (n; 0).

7. Let X₁, X2,..., Xn be independent variables, X; being N(µi, 1), and let y = = x² + x² +...+X²/2. Show that the characteristic function of Y is by(t) = 1 (1-2it)n/2 exp ite 1-2it where 0 = ²+²+ ...+. The random variables Y is said to have the non-central chi-squared distribution with n degrees of freedom and non-centrality parameter 0, written x² (n; 0).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
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7. Let X₁, X2,..., Xn be independent variables, X; being N(µi, 1), and let y = = x² + x² +...+ X² 2. Show that the characteristic function of Y is by(t) = 1 (1-2it)n/2 exp ite 1-2it where 0 = ²+²+ ...+. The random variables Y is said to have the non-central chi-squared distribution with n degrees of freedom and non-centrality parameter 0, written x² (n; 0).
Transcribed Image Text:7. Let X₁, X2,..., Xn be independent variables, X; being N(µi, 1), and let y = = x² + x² +...+ X² 2. Show that the characteristic function of Y is by(t) = 1 (1-2it)n/2 exp ite 1-2it where 0 = ²+²+ ...+. The random variables Y is said to have the non-central chi-squared distribution with n degrees of freedom and non-centrality parameter 0, written x² (n; 0).
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