HW2 - Mathematics Department, UCLA T. Richthammer winter...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Mathematics Department, UCLA T. Richthammer winter 09, sheet 2 Jan 09, 2009 Homework assignments: Math 170A Probability, Sec. 1 009. Let S be a sample space with probability law P . Let E, F be events with P ( E ) = 1 / 2, P ( F c ) = 2 / 3 and P ( E F ) = 2 / 3. Use the properties of a probability law to calculate P ( F ), P ( E F ) and P ( E - F ). Answer: P ( F ) = 1 - P ( F c ) = 1 / 3. P ( E F ) = P ( E ) + P ( F ) - P ( E F ) = 1 / 6. P ( E - F ) = P ( E ) - P ( E F ) = 1 / 3. 010. Let S be a sample space with probability law P . Use the properties of a probability law to show that for arbitrary events E 1 , E 2 we have (a) P ( E 1 E 2 ) = P ( E 1 ) + P ( E c 1 E 2 ) (b) P ( E 1 E 2 ) P ( E 1 ) + P ( E 2 ) (c) P (( E 1 - E 2 ) ( E 2 - E 1 )) = P ( E 1 ) + P ( E 2 ) - 2 P ( E 1 E 2 ) Answer: (a) This follows from the addition rule, as the set on the LHS is the disjoint union of the two sets on the RHS. (b) This follows from (a) as E c 1 E 2 E 2 , and so P ( E c 1 E 2 ) P ( E 2 ). (c) This follows by expressing all sets as disjoint unions of E 1 - E 2 , E 2 - E 1 and E 1 E 2 . (Draw a Venn diagram!) 011. Let S be a sample space with probability law P . Use the properties of a probability law to show that for arbitrary events E 1 , E 2 , E 3 we have (a) P ( E 1 E 2 E 3 ) = P ( E 1 ) + P ( E c 1 E 2 ) + P ( E c 1 E c 2 E 3 ) (Hint: Venn diagram.) (b) P ( E 1 E 2 E 3 ) P ( E 1 ) + P ( E 2 ) + P ( E 3 ) (Hint: Use (a)) (c) Generalize (a) and (b) to a union of n sets E 1 , . . . , E n and prove these formulas. Answer: (a) This follows from the addition rule, as the set on the LHS is the disjoint union of the two sets on the RHS. (b) This follows from (a) as E 2 E c 1 E 2 and E 3 E c 1 E c 2 E 3 . (c) P ( E 1 . . . E n ) = P ( E 1 ) + P ( E c 1 E 2 ) + . . . + P ( E c 1 . . . E c n - 1 E n ) P ( E 1 ) + P ( E 2 ) + . . . + P ( E n ). Proof similarly to (a) and (b). 012. Let S be a sample space with probability law P . Use the properties of a probability law to show that for arbitrary events E 1 , E 2 , E 3 we have (a) P ( E 1 E 2 ) P ( E 1 ) + P ( E 2 ) - 1 (b) P ( E 1 E 2 E 3 ) P ( E 1 ) + P ( E 2 ) + P ( E 3 ) - 2 (Hint: Use 011.(b).) Answer: (a) P ( E 1 E 2 ) = P ( E 1 ) + P ( E 2 ) - P ( E 1 E 2 ) P ( E 1 ) + P ( E 2 ) - 1 as P ( ... ) 1. (b) By 011.(b) P (( E 1 E 2 E 3 ) c ) = P ( E c 1 E c 2 E c 3 ) P ( E c 1 ) + P ( E c 2 ) + P ( E c 3 ). Using P ( E c ) = 1 - P ( E ) the result follows.
Background image of page 1

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

View Full DocumentRight Arrow Icon
013. Let S be a sample space with probability law P . Write down the inclusion-exclusion formula explicitely (i.e. without using the summation symbol ) for n = 2 and n = 3. Answer:
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/07/2010 for the course MATH 170A 170A taught by Professor Richthammer during the Winter '10 term at UCLA.

Page1 / 4

HW2 - Mathematics Department, UCLA T. Richthammer winter...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online