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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 , . . . ....
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 Fall '03
 aldous
 Probability, Probability theory, probability density function, density function, Jack, Cumulative distribution function, Jill

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