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Unformatted text preview: Midterm Examination Solutions, Statistics 220, Fall 2005 19 October 2005 1 Binomial vs. Poisson X is a binomial random variable, n = 10, p = 0 . 1. Y is a Poisson random variable, = 1. (a) What is E [ X ]? (1 pt) What is Var ( X )? (1 pt) What is the probability that X = 2? (3 pts) (b) What is E [ Y ]? (1 pt) What is Var ( Y )? (1 pt) What is the probability that Y = 2? (5 pts) (a) The expectation of a binomial is np = 1. The variance is np (1 p ) = 0 . 9. Pr ( X = 2) = 10 2 (0 . 1) 2 (0 . 9) 8 = 10 9 8! 2 8! (0 . 1) 2 (0 . 9) 8 = 45(0 . 1) 2 (0 . 9) 8 = 0 . 1937102 (b) The expectation of a Poisson is = 1. The variance is = 1. Pr (() Y = 2) = e 1 1 2 2! = 0 . 1839397 2 Camels Bactrian camels ( Camelus bactrianus ) are used as beasts of burden in Central Asia. A survey of the camels available in a certain town in Afghanistan shows that their mean weight is 1600 pounds, with a standard deviation of 150 pounds. (a) What are the mean and standard deviation of the camels weights in kilograms? (2 pts) Hint : 1 kg = 2.2 lb. (b) The camels are now loaded with water casks weighing exactly 120 kilograms each. What is the mean and standard deviation of their total weight? (3 pts) (a) Multiplying the measurements by a constant multiplies the mean and the standard deviation by the same constant. So the mean weight is 727 kg, and the standard deviation of the weight is 68 kg. 1 (b) Adding a constant increases the mean by that amount, but doesnt change the standard deviation. Mean loaded weight, 847 kg, standard deviation 68 kg. 3 Coins (a) What are the probabilities of getting the following sequences of heads and tails from 10 consecutive tosses of a fair coin? (2 pts) (i) HHHHHHHHHH (ii) HTHTHTHTHT (iii) HHTTTTTTHH (b) What is the probability of getting 0 heads in 10 consecutive tosses? 5 heads out of 10? 6 out of 10? (3 pts) (a) All three sequences are the same length, and every individual...
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 Fall '05
 Shalizi
 Statistics, Binomial

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