s6 - Math 4653 Elementary Probability Spring 2007...

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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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s6 - Math 4653 Elementary Probability Spring 2007...

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