Example Class 2

# Example Class 2 - 3 A player throws darts at a target On...

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The University of Hong Kong Department of Statistics and Actuarial Sciences STAT1801 Probability and Statistics: Foundations of Actuarial Sciences (10-11) Example Class 2 1. Urn 1 contains four red, three blue and two green balls while Urn 2 contains two red, three blue, and four green balls. A ball is drawn from urn 1 and put into urn 2. The balls in Urn 2 are then mixed, and one is drawn at random. (a) What is the probability that both balls drawn are red in color? (b) What is the probability that a blue ball is drawn from Urn 2? (c) Given that a green ball is drawn from Urn 2, what is the probability that a green ball is drawn from Urn 1? 2. In a certain city, 8% of all adults suffer from diabetes. A certain diagnostics gives a correct result with probability 95%. A person is randomly drawn and the diagnostics shows a positive result. What is the probability that the person diagnosed does not suffer from diabetes actually? 3. A player throws darts at a target. On each trial, independently of the other trials, he hits
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Unformatted text preview: 3. A player throws darts at a target. On each trial, independently of the other trials, he hits the bull's-eye with probability .05. How many times should he throw so that his probability of hitting the bull's-eye at least once is .5? 4. The following table shows a cumulative distribution function (cdf) of a discrete random variable: (a) Find the probability mass function (pmf). (b) Find f(3.3) and F(3.3) 5. Let ? ( ± ) = 1 − exp( −²± ³ ) , ´µ ± ≥ , ´µ ± < 0 , for some constants ² , ³ > 0 . (a) Show that ? ( ± ) is a proper cdf. (b) Find the probability density function (pdf). (c) Find ± such that ¶ ( · > ± ) = 0.1 . 6. Suppose that · has the density function µ ( ± ) = ¸±¹ −º± , ± ≥ , ± < 0 where º > 0 is a constant. (a) Find the value of ¸ . (b) Find the cdf. (c) Find ¶ (0 < · < 1 º ) . k 0 1 2 3 4 5 F(k) 0 .1 .3 .7 .8 1...
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