Let X_1, X_2,...X_(n_X) sim N(mu_X, sigma^2) and Y_1, Y_2,...,Y_(n_Y) sim N(mu_Y,sigma^2) where X_i and Y_j are independent for all i and j. Find the distribution of 1/sigma^2[(n_x-1)S_x^2]

sophiottebar

sophiottebar

Answered question

2022-12-18

Let X 1 , X 2 , . . . , X n x N ( μ x , σ 2 ) and Y 1 , Y 2 , . . . , Y n Y N ( μ Y , σ 2 ) where X i and Y j are independent for all i and j. Findthe distribution of
1 σ 2 [ ( n x 1 ) S x 2 + ( n y 1 ) S Y 2 ]

Answer & Explanation

Camryn Moreno

Camryn Moreno

Beginner2022-12-19Added 4 answers

Chi square distribution its defined as square of the standard normal distribution
( x μ σ ) 2 x 1 2 s x 2 = 1 m x 1 ( x i x ¯ ) 2
So, ( m x 1 ) s x 2 = ( x i x ¯ ) 2
Similiarly i = 1 n x ( r i μ ¯ ) 2 σ 2 x m x 1 2
X i s and r i s are independent thus sum of two chi square variate will also be ch-square
( x i x ¯ ) 2 σ 2 + ( x i x ¯ ) 2 σ 2 x m x + n x 2 2 1 σ 2 [ ( n x 1 ) x 2 ( n x 1 ) x 2 ] sin x n x 2 + n x 2

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