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Unformatted text preview: Math 42 Probability Practice Problems 1. [2004 Exam] When a certain farms crop is sprayed with insecticide, the amount of time that the crop actually stays free of bugs is a randomly varying quantity. The probability density function on this bug-free length of time is given by f ( t ) = Ct if 0 t 1 , C t 3 if t > 1 , otherwise, where t is measured in weeks, and C is a positive constant. (a) Find C , using the fact that f is a probability density function. (b) What is the probability that the crops bug-free time lasts no more than 3 weeks? (c) Find the mean amount of time that the crop is free of bugs. 2. [2005 Exam] For this problem, use the following information about any normal (bell-shaped or Gaussian) probability density function f : f has the general form f ( x ) = 1 2 e- ( x- ) 2 / 2 2 Z + - f ( x ) dx . 84 Z +2 - f ( x ) dx . 98 Suppose that a quality-assurance tester has determined that the amount of sodium in a bottle of Babbling Brook Spring Water is a random variable, having a normal distribution with mean 10 mg and standard deviation 1 mg. Using the above facts, find the probability that a randomly selected bottle of Babbling Brook contains between 8 and 9 mg of sodium. Justify your answer by writing an integral expression that represents this probability and showing how to evaluate this integral. 3. [2005 Exam] The age of a citizen in the country of Bishkadu is a random variable that can be closely modeled by the following probability density function: f ( t ) = C if 0 t 24 , Ce- ( t- 24) / 16 if t 24 , otherwise, where t is measured in years, and C is a positive constant....
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