exercise:Ef47114a42: Difference between revisions

Latest revision as of 03:37, 2 June 2024

Generalize Proposition as follows. For [math]i=1,\dots,d[/math] let [math]X_i\sim\mathcal{N}(\mu_i,\sigma_i)[/math] be independent Gaussian random variables. Let [math]\lambda_i\not=0[/math] be real numbers. Show that [math]X:=\lambda_1X_1+\cdots+\lambda_dX_d[/math] is again a Gaussian random variable with mean [math]\mu=(\mu_1+\cdots+\mu_d)/d[/math] and [math]\sigma^2=\lambda_1^2\sigma_1^2+\cdots+\lambda_d^2\sigma_d^2[/math].

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+Generalize [[guide:C64b6be05f#LK-GAUSS-LEM |Proposition]] as follows. For <math>i=1,\dots,d</math> let <math>X_i\sim\mathcal{N}(\mu_i,\sigma_i)</math> be independent Gaussian random variables. Let <math>\lambda_i\not=0</math> be real numbers. Show that <math>X:=\lambda_1X_1+\cdots+\lambda_dX_d</math> is again a Gaussian random variable with mean <math>\mu=(\mu_1+\cdots+\mu_d)/d</math> and <math>\sigma^2=\lambda_1^2\sigma_1^2+\cdots+\lambda_d^2\sigma_d^2</math>.
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-\label{SUM-GAUSS-PROB} Generalize [[#LK-GAUSS-LEM |Proposition]] as follows. For <math>i=1,\dots,d</math> let <math>X_i\sim\mathcal{N}(\mu_i,\sigma_i)</math> be independent Gaussian random variables. Let <math>\lambda_i\not=0</math> be real numbers. Show that <math>X:=\lambda_1X_1+\cdots+\lambda_dX_d</math> is again a Gaussian random variable with mean <math>\mu=(\mu_1+\cdots+\mu_d)/d</math> and <math>\sigma^2=\lambda_1^2\sigma_1^2+\cdots+\lambda_d^2\sigma_d^2</math>.