STAT 400 Lecture AL1 1 Examples for 3 3 Spring 2015 Dalpiaz At Initech the salaries of the employees are normally distributed with mean 36 000 and standard deviation 5 000 a Mr Smith is paid 42 000 What proportion of the employees of Initech are paid less that Mr Smith 42 000 36 000 P X 42 000 P Z P Z 1 20 0 8849 5 000 b What proportion of the employees have their salaries over 40 000 40 000 36 000 P X 40 000 P Z P Z 0 80 1 0 7881 0 2119 5 000 c Suppose 10 Initech employees are randomly and independently selected What is the probability that 3 of them have their salaries over 40 000 Let Y number of employees out of 10 who have salaries over 40 000 Then Y has Binomial distribution n 10 p 0 2119 see b P Y 3 10 C 3 0 2119 3 0 7881 7 0 2156 d What proportion of the employees have their salaries between 30 000 and 40 000 40 000 36 000 30 000 36 000 P 30 000 X 40 000 P Z 5 000 5 000 P 1 2 Z 0 80 0 7881 0 1151 0 6730 e Mrs Jones claims that her salary is high enough to just put her among the highest paid 15 of all employees working at Initech Find her salary Need x such that P X x 0 15 area to the right is 0 15 First need z such that P Z z 0 15 z 1 04 X Z f x 36 000 5 000 1 04 41 200 Ms Green claims that her salary is so low that 90 of the employees make more than she does Find her salary Need x such that P X x 0 90 area to the right is 0 90 First need z such that P Z z 0 90 z 1 28 X Z 2 x 36 000 5 000 1 28 29 600 Suppose that the lifetime of Outlast batteries is normally distributed with mean 240 hours and unknown standard deviation Suppose also that 20 of the batteries last less than 219 hours Find the standard deviation of the distribution of the lifetimes Need Know P X 219 0 20 First need z such that P Z z 0 20 z 0 84 X Z 219 240 0 84 21 0 84 25 hours Let X be normally distributed with mean and standard deviation Then the moment generating function of X is e t MX t MX t E etX e t z 1 2 e t since et x e 1 2 z 2 2 2 2 t e x 2 2 2 2 dx 2 2 dz e t t 2 1 2 e z t 2 2 2 1 2 t 2 e z t 2 2 is the probability density function of a N t 1 random variable Let Y a X b Then M Y t e b t M X a t Therefore Y is normally distributed with mean a b and variance a 2 2 standard deviation a 1 continued g All Initech employees receive a memo instructing them to put away 4 of their salaries plus 100 per month 1 200 per year in a special savings account to supplement Social Security What proportion of the employees would put away more than 3 000 per year Y 0 04 X 1 200 Y 3 000 P Y 3 000 X 45 000 45 000 36 000 P X 45 000 P Z P Z 1 80 1 0 9641 0 0359 5 000 2 dz OR Y 0 04 36 000 1 200 2 640 Y 0 04 5 000 200 3 000 2 640 P Y 3 000 P Z P Z 1 80 1 0 9641 0 0359 200 3 Suppose the average daily temperature in degrees Fahrenheit in June in Anytown is a random variable T with mean T 85 and standard deviation T 7 The daily air conditioning cost Q in dollars for Anytown State University is related to T by Q 120 T 750 Suppose that T is a normal random variable Compute the probability that the daily air conditioning cost on a typical June day for the university will exceed 12 210 Q has Normal distribution Q 120 T 750 120 85 750 10 950 Q2 120 2 T2 120 2 7 2 840 2 Q 840 12 210 10 950 P Q 12 210 P Z P Z 1 50 1 1 50 1 0 9332 0 0668 840 OR 12 210 120 T 750 P Q 12 210 P T 95 5 95 5 85 P Z 7 P Z 1 50 1 1 50 1 0 9332 0 0668 T 95 5 4 Consider a random variable X with the moment generating function MX t a 2 e 3 t 8 t exp 3 t 8 t 2 Find P X 0 Normal distribution MX t e t 2 2 t 2 2 X has a Normal distribution with 3 and 2 8 E X 3 Var X 2 16 4 0 3 P X 0 P Z P Z 0 75 1 0 75 1 0 2266 0 7734 4 a Find P 1 X 9 9 3 1 3 P 1 X 9 P Z P 1 00 Z 1 50 4 4 1 50 1 00 0 9332 0 1587 0 7745 EXCEL NORMSDIST z gives P Z z z NORMSINV p gives z such that P Z z p NORMDIST x 1 gives P X x NORMDIST x 0 gives f x p d f of N 2 NORMINV p gives x such that P X x p where X is N 2 where X is N 2 5 Show that the odd moments of N 0 2 are zero and the even moments are 2n 2 n 2 n 2 n n Taylor Formula MX t tr M r 0 r 0 r tr tr X r E r r r r 0 r 0 Since X is N 0 2 2 n t 2n 2 t 2 M X t exp n 2 n 0 2 n Therefore if r is odd r 0 if r 2 n is even 2n 1 r 2 n n r OR Def x u x 1 e u du x 0 0 x x 1 x 1 n n 1 1 2 if n is an integer 2n 2 n 2 n 2 n n 2n x 2n u 1 2 e x n 2n 2 0 dx x 2 n 0 x2 2 2 2 2 2 du 1 x dx 2 u n 1 2 e u 2 2 dx 1 2n 1 2n 3 2n 5 3 1 2n 2 n 1 2 n 2n 2n 2 2n 4 4 2 2 n 1 n n 2 2 n 2 n 2 n 2n 2 n n 2 2 2 dx du 2u du 2 …
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