This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 4653: Elementary Probability: Spring 2007 Homework #6. Problems and Solutions 1. Sec. 4.2: #6: A Geiger counter is recording background radiation at an average rate of one hit per minute. Let T 3 be the time in minutes when the third hit occurs after the counter is switched on. Find P (2 ≤ T 3 ≤ 4). Solution. The random variable T 3 has a gamma distribution with parameter r = 3 and λ = 1, which gives us a density function f ( t ) = λ r t r 1 Γ( r ) t r 1 e λt = 1 2 t 2 e t . Therefore, P (2 ≤ T 3 ≤ 4) = Z 4 2 t 2 e t 2 dt = e t t 2 2 + t + 1 4 2 = e 4 (8 + 4 + 1) + e 2 (2 + 2 + 1) = 5 e 2 13 e 4 ≈ . 4386 . 2. Sec. 4.4: #4: Suppose X has uniform distribution on ( 1 , 1). Find the density of Y = X 2 . Solution. It is easy to show that  X  has uniform distribution on [0 , 1). Then Y has density f Y ( y ) = f  X   dy/d  x  = 1 2  x  = 1 2 √ y for 0 < y < 1 , otherwise . 3. Sec. 4.5: #2: Find the cumulative distribution functions of: a) the binomial (3 , 1 / 2) distribution; b) the geometric (1 / 2) distribution on { 1 , 2 , . . . } . Solution. a ) P ( X ≤ x ) = for x < , 1 / 8 for 0 ≤ x < 1 , 1 / 2 for 1 ≤ x < 2 , 7 / 8 for 2 ≤ x < 3 , 1 for x ≥ 3 . b ) P ( X ≤ x ) = for x < 1 , 2 k 1 2 k for k ≤ x < k + 1 , k = 1 , 2 , . . . ....
View
Full
Document
This note was uploaded on 09/14/2009 for the course STAT 134 taught by Professor Aldous during the Fall '03 term at Berkeley.
 Fall '03
 aldous
 Probability

Click to edit the document details