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Example Class 8

Example Class 8 - 525 hours 3 A die is continually rolled...

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The University of Hong Kong Department of Statistics and Actuarial Science STAT1801 Probability and Statistics: Foundations of Actuarial Science (10-11) Example Class 8 1. Suppose 𝑋𝑋 1 , 𝑋𝑋 2 , … , 𝑋𝑋 𝑛𝑛 are i.i.d., each with some unknown distribution that has an expected value, 𝜇𝜇 , equals to 25.7 and a standard deviation, 𝜎𝜎 , equals to 4.3. If 𝑛𝑛 = 50 , find the approximate probability that the sample mean is less than 25. Repeat for 𝑛𝑛 = 100 . 2. There are 100 light bulbs whose lifetimes are independently distributed as exponential with mean 5 hours. If the bulbs are used one at a time, with a failed bulb being replaced immediately by a new one, what is the probability that there is still a working bulb after 525 hours? 3. A die is continually rolled until the total sum of all rolls exceeds 300. What is the
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Unformatted text preview: 525 hours? 3. A die is continually rolled until the total sum of all rolls exceeds 300. What is the probability that at least 80 rolls are necessary? 4. Let ? 1 , ? 2 , … , ? ± be a random sample of size ± from the exponential distribution whose pdf is given by ´ ( µ ) = ¶· −¶µ , µ > 0, ¶ > 0 a) Show that the sample mean ? is an unbiased estimator of 1 ¶ . b) Given a random sample of size 5 : 3.5, 8.1, 0.9, 4.4, 0.5 , find a point estimate of ¶ . 5. Let ? ¸ , ¸ = 1,2, … , ± be Bernoulli random variables such that ? ¸ = ¹ 1 , º , 1 − º a) Show that ? is an unbiased estimator of º . b) Find an unbiased estimator of the variance of ? ....
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