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7.1. The Binomial Probability Distribution Example: X2 : representing the number of heads as flipping a fair coin twice. H : head X2 0 X2 1 X2 2 □ □ T T □ □ H T T H □ □ H H T : tail . P( X 2 0) 1 1 2 1 1 2 2 0 2 2 (1 combination) 1 1 2 1 1 (2 combinations) P( X 2 1) 2 2 2 1 2 2 P( X 2 2) 1 1 2 1 1 2 2 2 2 2 (1 combination) 2 1 P( X 2 i) f 2 (i) i 2 (number of combinatio ns) ( the probabilit y of every combinatio n) 2 , i 0, 1, 2. X3 : representing the number of heads as flipping a fair coin 3 times. X3 0 □ □ □ T T T P( X 3 0) (1 combination) 2 2 2 0 2 1 1 1 1 3 1 3 X3 1 X3 2 X3 3 □ □ □ H T T T H 1 1 1 3 1 T P( X 3 1) 3 2 2 2 1 2 T T H □ □ □ H H T H T 1 1 1 3 1 H P( X 3 2) 3 2 2 2 2 2 T H H □ □ □ H H H P( X 3 3) (1 combination) 2 2 2 3 2 1 1 1 3 1 3 (3 combinations) 3 (3 combinations) 3 3 1 P( X 3 i) f 3 (i) i 2 (number of combinatio ns) ( the probabilit y of every combinatio n) 3 , i 0, 1, 2, 3. Xn : representing the number of heads as flipping a fair coin n times. Then, n □ □ …………… □ n 1 1 T…………….T P( X n 0) 2 0 2 n Xn 0 T 2 n (1 combination) n □ □ …………… □ H T……..……..T n 1 1 T P( X n 1) n 2 1 2 n n Xn 1 T H……… n (n combinations) T T …………..H n 1 P( X n i) f n (i) i 2 (number of combinatio ns) (the probabilit y of every combinatio n) n Note: the number of combinations is equivalent to the number of ways as drawing i balls (heads) from n balls (n flips). Example: Z3 : representing the number of successes over 3 trials. S : Success F : Failure 3 Suppose the probability of the success is 1 2 while the probability of failure is . 3 3 Then, □ □ □ F 3 1 1 2 2 F P(Z 3 0) 3 3 0 3 3 0 Z3 0 F 3 0 3 (1 combination) □ □ □ S F F F S 3 1 2 1 2 F P(Z 3 1) 3 3 3 1 3 3 2 Z3 1 2 (3 combinations) F F S □ □ □ S S F F 1 2 3 1 2 S P(Z 3 2) 3 3 3 2 3 3 2 Z3 2 S 2 (3 combinations) F S S □ □ □ S 3 1 1 2 2 S P(Z 3 3) 3 3 3 3 3 3 Z3 3 S 0 3 0 (1 combination) 3i 3 1 2 P( Z 3 i) f 3 (i) i 3 3 (number of combinatio ns) ( the probabilit y of every combinatio n) i 4 , i 0, 1, 2, 3. Zn : representing the number of successes over n trials. Then, n □ □ …………… □ n 1 2 F F…………….F P(Z n 0) 3 0 3 n Zn 0 0 2 3 n (1 combination) n □ □ ……… □ S F…….. .F Zn 1 n F S……… F 1 2 P( Z n 1) n 3 3 n 1 n 1 2 1 3 3 n 1 (n combinations) F F … .S n i n 1 2 P( Z n i) f n (i) i 3 3 (number of combinatio ns) (the probabilit y of every combinatio n) i 5 From the above example, we readily describe the binomial experiment. Properties of Binomial Experiment X: representing the number of successes over n independent identical trials. The probability of a success in a trial is p while the probability of a failure is (1-p). Binomail Probability Distribution: Let X be the random variable representing the number of successes of a Binomial experiment. Then, the probability distribution function for X is n n! i n i n i P( X i) f x (i) p i 1 p p 1 p , i 0,1, 2,, n . i!n i ! i Properties of Binomial Probability Distribution: A random variable X has the binomial probability distribution f (x) with parameter p , then E ( X ) np and Var ( X ) np(1 p) . [Derivation:] 6 n n i n! n i n i E ( X ) i f (i ) i p 1 p i p i 1 p i!n i ! i 0 i 0 i 0 i n n n! n! n i n i i i p 1 p p i 1 p i!n i ! i 1 i 1 i 1! n i ! n n n 1! ( n 1) ( i 1) p i 1 1 p i 1!n 1 i 1! i 1 n 1 n 1! p j 1 p n1 j ( j i 1) np j 0 j!n 1 j ! n 1! p j 1 p n1 j is the probabilit y np (since j!n 1 j ! n np distributi on of a binomial random variable over n 1 trials) The derivation of Var ( X ) np(1 p) is left as exercise. How to obtain the binomail probability distribution: (a) Using table of Binomail distribution. (b) Using computer by some software, for example, Excel or Minitab. Online Exercise: Exercise 7.1.1 Exercise 7.1.2 7