binomexamp

binomexamp - More Examples: Binomial, Hypergeometric,...

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More Examples: Binomial, Hypergeometric, Negative Binomial Example 1 When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Let X = the number of defective boards in a random sample of size n = 25 . a. Determine P ( X · 2) , P ( X ¸ 5) , and P (1 · X · 4) . Solution X » Bin (25 ; 0 : 05) , so: P ( X · 2) = B (2;25 ; 0 : 05) = 0 : 873 P ( X ¸ 5) = 1 ¡ P ( X · 4) = 1 ¡ B (4;25 ; 0 : 05) = 1 ¡ 0 : 993 = 0 : 007 P (1 · X · 4) = P ( X · 4) ¡ P ( X · 0) = B (4;25 ; 0 : 05) ¡ B (0;25 ; 0 : 05) = 0 : 993 ¡ 0 : 277 = 0 : 716 : b. What is the probability that none of the 25 boards are defective? Solution P ( X = 0) = P ( X · 0) = 0 : 277 : c. Calculate the expected value and standard deviation of X . Solution The expected value and standard deviation of X are: E ( X ) = np = (25)(0 : 05) = 1 : 25 ¾ X = q np (1 ¡ p ) = q (25)(0 : 05)(0 : 95) = 1 : 1875 : Example 2 Suppose that only 20% of all drivers come to a complete stop at an intersection having ‡ashing red lights in all directions when no other cars are visible. What is the probability that, of 20 randomly chosen drivers coming to an intersection under these conditions: a. At most …ve will come to a complete stop? Solution Let X = number of cars that come to a complete stop. Then X » Bin (20 ; 0 : 20) . Therefore P ( X · 5) = B (5;20 ; 0 : 20) = 0 : 804 . b. Exactly …ve will come to a complete stop? Solution P ( X = 5) = b (5;20 ; 0 : 20) = B (5;20 ; 0 : 20) ¡ B (4;20 ; 0 : 20) = 0 : 174 : c. At least …ve will come to a complete stop? Solution P ( X ¸ 5) = 1 ¡ P ( X · 4) = 1 ¡ B (4;20 ; 0 : 20) = 0 : 370 : d. How many of the next 20 drivers do you expect to come to a complete stop? Solution E ( X ) = np = (20)(0 : 20) = 4 : 1
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Example 3 Prove that E ( X ) = np and V ( X ) = np (1 ¡ p ) , where X is a binomial random variable. Solution Let X » Bin ( n;p ) . Then: E ( X ) = n X x =0 x à n x ! p x (1 ¡ p ) n ¡ x = n X x =1 x n ! x
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binomexamp - More Examples: Binomial, Hypergeometric,...

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