Final_Exam_Solution

# Final_Exam_Solution - Final Exam Solution ORIE 3500/5500...

This preview shows pages 1–2. Sign up to view the full content.

Final Exam Solution ORIE 3500/5500: Engineering Probability and Statistics II Question 1 (i) (a) Probability Law: For a statistical experiment with sample space Ω and collection of events F , a probability law is a function P : F → R such that i. P ( A ) 0 for any event A , ii. P (Ω) = 1, and iii. P ( A 1 A 2 ... ) = P ( A 1 ) + P ( A 2 ) + ··· for disjoint events A 1 ,A 2 ,... . (b) Estimator: If X 1 ,...,X n are the available data to estimate an unknown parameter θ , then a function g ( X 1 ,...,X n ) is an estimator for θ . (c) Unbiased estimator: An estimator ˆ θ is unbiased for the parameter θ if E θ ( ˆ θ ) = θ θ. (d) Maximum likelihood estimator: If X 1 ,...,X n are the data with joint PDF (or PMF) L ( θ ) = f X ( x 1 ,...,x n ), then the MLE is an estimator ˆ θ which maximizes L ( θ ) over θ . (ii) (a) M Z ( s ) = E [ e s ( a ( X - 1)+ bY ) ] = E [ e saX e - sa e sbY ] = e - sa E [ e saX ] E [ e sbY ] = e - sa M X ( sa ) M Y ( sb ) . (b) ˆ

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

Final_Exam_Solution - Final Exam Solution ORIE 3500/5500...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online