stats150-5

# stats150-5 - we see the ﬁrst kurrawong 4 c Let U be the...

This preview shows pages 1–7. Sign up to view the full content.

STAT 150 MIDTERM - SPRING 2007 Instructor: Steven Evans Instructions: Each question part is worth an equal amount. Explain carefully and completely what you are doing at each step in full sen- tences. Quote any results that you use from class. NAME: SID: 1)a) 1)b) 1)c) 2)a) 2)b) 2)c) TOTAL 1

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

View Full Document
2 1) Two species of birds, kookaburras and kurrawongs, are sitting along a barbed-wire fence somewhere in Australia. We travel east along the fence from a certain ﬁxed point that we call position 0 and write L ( t ) for the number of kookaburras we have seen after traveling t miles. Similarly, let M ( t ) be the number of kurrawongs we have seen after traveling t miles. Suppose that L and M are independent homogeneous Poisson processes with intensities λ and μ . a) For n = 1 , 2 ,... ﬁnd the distribution of the distance (in miles) that we have to travel along the fence from position 0 until we see the n th bird.
3 b) Find the distribution of the number of kookaburras we see before

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

View Full Document

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.

Unformatted text preview: we see the ﬁrst kurrawong. 4 c) Let U be the distance (in miles) from 0 to the ﬁrst bird we see and V be the distance (in miles) from 0 to the ﬁrst bird of the other species (that is, if the ﬁrst bird we see is a kookaburra, then V is the distance to the ﬁrst kurrawong and vice versa ). Find the distribution of V-U . 5 2) Consider the discrete time Markov chain X ,X 1 ,X 2 ,... with state-space { , 1 , 2 , 3 , 4 } and transition matrix 1 / 2 1 / 2 1 / 3 2 / 3 1 / 5 1 / 5 1 / 5 1 / 5 1 / 5 2 / 3 1 / 3 1 a) Which states are persistent and which states are transient? 6 b) What is the approximate value of P { X n = 0 | X = 2 } when n is large? 7 c) Let T = min { n ≥ 1 : X n = 1 } . What is E [ T | X = 1]?...
View Full Document

{[ snackBarMessage ]}

### Page1 / 7

stats150-5 - we see the ﬁrst kurrawong 4 c Let U be the...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online