Statistics Homework Solutions 79

# Statistics Homework Solutions 79 - 79 SECTION 4.3 0 2240 Â...

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Unformatted text preview: 79 SECTION 4.3 0 2240 Â¡ Â¡ Â¡ Â¤ Â¥ PX 32 2 e3 e3 0! 1! 2! 0 049787 0 14936 0 22404 Â¢ Â¨ Â¢ Â¢ Â¨ Â¢ Â¨ Â¡ 0 4232 Â¨ Â¤ Â¢Â¥ PX 33 3 e3 2! 3! 0 22404 0 22404 Â¢ Â¨ 3 2 Â¨ Â¢ Â¡ e Â¡ Â¢ Â¤ Â¡ 1 73. Â¢ Â¨ Â¨ 3. Â¡ Â¤ Â¡ Â¥Â¤ Poisson 3 , ÏƒX 3 Â¤ Â¨ Â¡ Poisson 3 , ÂµX Â¡ Â¡ Â¡ 0 5974 Â¨ Â¥ 1! 0 14936 32 Â¡ 3 PX Â¢Â¥ e 1 31 Â¤ PX Â¡ Â¡ 4 Â¥ Â¤ Â¨ Â¥Â¤ Let X be the number of messages that fail to reach the base station. Then X is the number of successes in n 1000 Bernoulli trials, each of which has success probability p 0 005. The mean of X is np 1000 0 005 5. Since n is large and p is small, X Poisson 5 to a very close approximation. Â¡ Â¡ Â¡Â¥ Â¥Â¤ Â¨ Â¤ Â¦Â¥ Â¤ Â¡ 55 e 3 3! 0 14037 Â¡ Â¡Â¥ 3 Â¨ Â¡ (a) P X Â¨ Â¤ (b) The event that fewer than 994 messages reach the base station is the same as the event that more than 6 messages fail to reach the base station, or equivalently, that X 6. 3 e 3! 0 17547 Â¨ Â¡ 2! 0 14037 55 Â¡ e 55 4 e 55 5 e 55 6 4! 5! 6! 0 17547 0 14622 0 2378 Â¨ 2 Â¡ 55 Â¨ Â¨ Â¡ Â¡ Â¡ 0! 1! 0 03369 0 08422 e Â¡ 1 Â¨ 55 Â¡ e 0 Â¡ Â¨ Â¡ Â¨ Â¡ Â¡Â¥ Â¨ Â¤ Â¨ Â¡ Â¡ Â¡Â¥ Â¡ Â¤ Â¡ 2 2361 55 Â¨ Â¡ 5 e Â¡ Â¤ Â¡ Â¡ (d) ÏƒX 0 00674 5 1 1 6 Â¡ (c) ÂµX PX 1 Â¡ 6 PX Â¡ 5. Â¢Â¥ (e) Since X Â¤ (d) Since X 1 Â¡ Â¤ X 31 Â¢Â¥ Â¡ (c) P 1 3 PX Â¤ e 0 30 Â¡ PX Â¥ Â¡Â¥ 2 3 3! (b) P X 33 e Â¡ 3 Â¨ (a) P X ...
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