Write the bivariate normal pdf f (x, y; ?1, ?2, ?3, ?4,

QUESTION:

Write the bivariate normal pdf \(f\left(x, y ; \theta_{1}, \theta_{2^{\prime}}, \theta_{3^{\prime}} \theta_{4}, \theta_{5}\right)\) in exponential form and show that \(Z_{1}=\sum_{i=1}^{n} X_{i}^{2}, Z_{2}=\sum_{i=1}^{n} Y_{i}^{2}, \quad Z_{3}=\sum_{i=1}^{n} X_{i} Y_{i^{\prime}} \quad Z_{4}=\sum_{i=1}^{n} X_{i}\), and \(Z_{5}=\sum_{i=1}^{n} Y_{i}\) are joint sufficient statistics for\(\theta_{1}, \theta_{2^{\prime}} \theta_{3} \theta_{4}\), and \(\theta_{5}\)

Equation Transcription:

, 

 ,

Text Transcription:

 f(x,y;theta_1,theta_2,theta_3,theta_4,theta_5

Z_1=sum_i=1^n X_i^2, Z_2=sum_i=1^n Y_i^2,   Z_3=sum_i=1^n X_i Y_i,    Z_4=sum_i=1^n X_i

Z_5=sum_i=1^n Y_i

theta_1,theta_2,theta_3,theta_4

theta_5