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Introduction A probability distribution is obtained when probability values are assigned to all possible numerical values of a random variable. It may also be denoted by the symbol f(x), in the continuous, which indicates that a mathematical function is involved. The sum of the probabilities for all the possible numerical events must equal 1.0. 1 Definition 6.1 A continuous random variable X is said to have a normal distribution with parameters and 2 , where and 2 0, if the pdf of X is 1 f ( x) e 2 1 x 2 2 x If X ~ N ( , 2 ) then E X and V X 2 2 6.2 Standard Normal Probability Distribution 2 The normal distribution with parameters 0 and 1 is called a standard normal distribution. A random variable that has a standard normal distribution is called a standard normal random variable and will be denoted by Z ~ N (0,1) . 3 Standardizing A Normal Distribution If X is a normal random variable with E ( X ) and V ( X ) 2 , the random variable X Z is a normal random variable with E ( Z ) 0 and V ( Z ) 1. That is Z is a standard normal random variable. 4 Determine the probability or area for the portions of the Normal distribution described. a) P (0 Z 0.45) b) P (2.02 Z 0) c) P ( Z 0.87) d) P (2.1 Z 3.11) e) P (1.5 Z 2.55) 5 a) P(0 Z 0.45) P( Z 0.45) P( Z 0) 0.67364 0.5 = 0.1736 b) P(2.02 Z 0) P(0 Z 2.02) = P( Z 2.02) P ( Z 0) 0.97831 0.5 = 0.47831 c) P( Z 0.87) 0.80785 6 d) e) 7 Determine Z such that a) P( Z Z ) 0.25 b) P( Z Z ) 0.36 c) P( Z Z ) 0.983 d) P( Z Z ) 0.89 8 a) P ( Z 0.6745) 0.25 b) P ( Z 0.3585) 0.36 c) P ( Z 2.1201) 0.983 d) P ( Z 1.2265) 0.89 9 Suppose X is a normal distribution N(25,25). Find a) P(24 X 35) b) P( X 20) 10 35 25 24 25 a) P(24 X 35) P Z 5 5 P(0.2 Z 2) = P( Z 2) P( Z 0.2) =P( Z 2) P( Z 0.2) =0.97725 0.42074 0.55651 20 25 b) P( X 20) P Z 5 P( Z 1) P( Z 1) 0.84134 11 When the number of observations or trials n in a binomial experiment is relatively large, the normal probability distribution can be used to approximate binomial probabilities. A convenient rule is that such approximation is acceptable when n 30, and both np 5 and nq 5. Definition 6.2 Given a random variable X ~ b(n, p), if n 30 and both np 5 and nq 5, then X ~ N ( np, npq) X np with Z npq 12 The continuous correction factor needs to be made when a continuous curve is being used to approximate discrete probability distributions. 0.5 is added or subtracted as a continuous correction factor according to the form of the probability statement as follows: c .c a) P( X x) P( x 0.5 X x 0.5) c .c b) P( X x) P( X x 0.5) c .c c) P( X x) P( X x 0.5) c .c d) P( X x) P( X x 0.5) c .c e) P( X x) P( X x 0.5) c.c continuous correction factor 13 Example 6.4 In a certain country, 45% of registered voters are male. If 300 registered voters from that country are selected at random, find the probability that at least 155 are males. 14 Solutions X is the number of male voters. X ~ b(300, 0.45) c .c P ( X 155) P( X 155 0.5) P( X 154.5) np 300(0.45) 135 5 nq 300(0.55) 165 5 154.5 300(0.45) 154.5 135 PZ P Z 300(0.45)(0.55) 74.25 P( Z 2.26) 0.01191 15 When the mean of a Poisson distribution is relatively large, the normal probability distribution can be used to approximate Poisson probabilities. A convenient rule is that such approximation is acceptable when 10. Definition 6.3 Given a random variable X ~ Po ( ), if 10, then X ~ N ( , ) with Z X 16 Example 6.5 A grocery store has an ATM machine inside. An average of 5 customers per hour comes to use the machine. What is the probability that more than 30 customers come to use the machine between 8.00 am and 5.00 pm? 17 Solutions X is the number of customers come to use the ATM machine in 9 hours. X ~ Po (45) 45 10 X ~ N (45, 45) c .c P( X 30) P ( X 30 0.5) P ( X 30.5) 30.5 45 PZ P( Z 2.16) 45 0.98461 18