# ps12_sol - Massachusetts Institute of Technology Department...

• Notes
• 3
• 100% (1) 1 out of 1 people found this document helpful

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

�� �� Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Solutions for Problem Set 12: Topic: Central Limit Theorem Due: No due date 1. It is not easy to calculate the CDF of the total weight and the desired probability, but an approximate answer can be quickly obtained using the central limit theorem. We want to calculate P ( S 102 > 200), where S 102 is the sum of the weights of 102 pretzels. The mean weight of a single pretzel is µ W = 2 . The variance of the weight of a single pretzel is 2 σ 2 = E [ W 2 ] µ W W = w 2 f W ( w ) dw 4 0 �� 2 3 = w 2 ( w 1) dw + w 2 (3 w ) dw 4 1 2 25 = 6 4 1 = 6 . Thus we have P ( S 102 > 200) = 1 P ( S 102 200) S 102 102 · 2 200 102 · 2 = 1 P 102 · (1 / 6) 102 · (1 / 6) 1 Φ 4 17 4 = 1 1 Φ 17 = Φ(0 . 9701) 0 . 8340 . 2. The probability that you will believe the fair coin to be biased is the probability that the fair coin will come up with more than 525 heads out of the 1000 tosses. Let S be the number of times the coin comes

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

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

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern