STAT 400 Discussion 6 Answers Spring 2015 1 The weight of fish in Lake Paradise follows a normal distribution with mean of 8 1 lbs and standard deviation of 2 5 lbs a What proportion of fish are between 9 lbs and 12 lbs 12 8 1 9 8 1 P 9 X 12 P Z P 0 36 X 1 56 2 5 2 5 0 9406 0 6406 0 3 b Mr Statman boasts that he once caught a fish that was just big enough to be in the top 2 5 of the fish population How much did his fish weigh Want x such that P X x 0 025 P Z 1 96 0 0250 c x 8 1 2 5 1 96 13 lbs If one catches a fish from the bottom 20 of the population the fish must be returned to the lake What is the weight of the smallest fish that one can keep Want x such that P X x 0 20 P Z 0 84 0 2005 0 20 x 8 1 2 5 0 84 6 lbs 2 Bob sells thingamabobs His yearly salary is 27 000 plus a commission of 10 of his sales His yearly sales are normally distributed with mean 100 000 and standard deviation 20 000 a Find the probability that Bob earns over 40 000 in a given year Let X denote Bob s yearly sales Let Y denote the amount Bob earns in a given year Then Y 0 10 X 27 000 Since X has a Normal distribution with mean 100 000 and standard deviation 20 000 Y also has a Normal distribution with mean 0 10 100 000 27 000 37 000 and standard deviation 0 10 20 000 2 000 40 000 37 000 P Y 40 000 P Z P Z 1 50 0 0668 2 000 b Find the missing value With probability 67 Bob earns over in a given year Need y such that P Y y 0 67 Find z such that P Z z 0 67 The area to the left is 0 33 z Using the standard normal table x z z 0 44 x 37 000 2 000 0 44 36 120 3 4 Let X be a continuous random variable with the probability density function c x 1 x 3 f x 0 3 otherwise Find the value of c that makes f x a valid probability density function a Must have f x dx 1 0 1 3 c x dx c x dx 1 b 0 1 5 c c Find the probability P X 1 3 P X 1 1 x 5 dx 0 80 1 9 c c 5 c 2 2 1 0 20 5 c Find the median of the probability distribution of X F X x 0 x 1 x FX x 1 x 2 y dy 5 10 1 0 FX x y 5 dy 1 1 10 0 x 3 0 median 3 1 median 2 1 10 10 2 x f x dx e 0 1 x2 10 10 median 2 Find the mean of the probability distribution of X d E X 5 dy F X median 4 x y x 3 F X x 1 FX 0 1 x 0 x2 5 1 0 3 dx 0 2 x 5 dx 1 27 26 1 73333 15 15 15 Find the variance of the probability distribution of X x3 E X x f x dx 5 1 2 0 2 2 Var X E X E X 2 3 dx 0 41 26 10 15 x3 5 2 dx 1 81 41 10 20 20 493 1 09556 450 P k a k k 2 3 4 5 6 5 Suppose S 2 3 4 5 6 and a Find the value of a that makes this is a valid probability model Must have P x 1 all x 1 ak k 2 first term a2 1 a 1 base a 2 a 1 0 a 1 5 2 1 5 0 2 a 5 1 0 618034 2 a Note where b is the golden ratio Find P outcome is divisible by 2 P outcome is divisible by 2 P 2 P 4 P 6 P 8 a2 a4 a6 a8 a2 1 a 2 first term 1 base a2 a 0 618034 a 1 1 6 Suppose a discrete random variable X has the following probability distribution f k P X k a k k 2 3 4 5 6 where a 1 0 618034 where a is the golden ratio Find the moment generating function of X M X t For which values of t does it exist MX t E et X etk a k k 2 a e t k 2 t a e 1 b zero otherwise k a 2 e 2t first term 1 base 1 a e t t ln 1 a ln 0 48121 Find E X 2 a 2 e 2 t 1 a e t a 2 e 2 t a e t 2 a 2 e 2 t a 3 e 3t M X t 2 2 1 a e t 1 a e t t ln E X M X 0 2a2 a3 1 a 2 2 a 3 a 3 618034 1 a OR E X 2 a 2 3 a 3 4 a 4 5 a 5 6 a 6 a E X 2 a 3 3 a 4 4 a 5 5 a 6 1 a E X Therefore E X a 2 a 2 a 3 a 4 a 5 a 6 a 2 1 a2 1 1 2 a 1 3 a 3 618034 1 a 1 a 1 a 1 a 7 A computer independently generates seven random numbers from a Uniform 0 1 distribution a What is the probability that exactly three will be in the interval from to 1 Let X denote the number of random numbers out of 7 that are between and 1 Then X has a Binomial distribution n 7 p 0 50 P X 3 7 C 3 0 50 3 0 50 4 0 2734375 b What is the probability that fewer than three will be in the interval from to 1 Let X denote the number of random numbers out of 7 that are between and 1 Then X has a Binomial distribution n 7 p 0 25 P X 3 7 C 0 0 25 0 0 75 7 7 C 1 0 25 1 0 75 6 7 C 2 0 25 2 0 75 5 0 1334839 0 3114624 0 3114624 0 7564087 8 For a statistics homework an instructor will randomly select 3 questions out of 10 to be graded Alex did not have enough time to prepare answers for all 10 questions he prepared answers to only 6 That is there are 6 good questions and 4 bad questions What is the probability that Alex has answers prepared for at least 2 out of the 3 questions selected by …
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