STAT 400 Lecture AL1 1 Spring 2015 Dalpiaz Answers for 3 1 Let X be a continuous random variable with the probability density function a f x k x 0 x 4 f x 0 otherwise What must the value of k be so that f x is a probability density function f x 0 1 f x dx 1 2 1 4 f x dx b k 4 x dx k x 0 2 3 k x 3 4 2 0 1 2 dx 0 k 16 3 k 3 0 1875 16 Find the cumulative distribution function of X F X x P X x x FX x P X x f y dy x 0 F X x 0 x 0 x 4 FX x 16 3 0 x 4 F X x 1 y dy 3 2 32 x y 16 3 0 1 8 x 3 2 c Find the probability P 1 X 2 2 P 1 X 2 f x dx 1 2 1 3 16 x dx 3 2 32 2 x 16 3 1 0 22855 OR P 1 X 2 F X 2 F X 1 d 1 2 3 2 8 1 1 3 2 0 22855 8 Find the median of the distribution of X That is find m such that P X m P X m 1 2 1 2 m f x dx 0 3 4 m 2 e m 3 16 m 3 x dx 3 m 3 3 2 1 x 2 m 2 16 3 8 0 4 2 2 51984 Find the 30th percentile of the distribution of X That is find a such that P X a 0 30 0 30 a a 0 f x dx 3 2 4 a 2 3 16 a x dx 3 3 a 3 2 3 1 x 2 a 2 16 3 8 0 2 4 2 1 79256 f Find X E X E X X g 4 0 x f x dx 3 x 16 4 3 3 x dx x 2 dx 16 0 5 4 3 2 x 2 2 4 16 5 0 Find X SD X 4 3 5 2 2 Var X X x f x dx X x 2 dx 2 4 2 16 0 2 3 2 72 4 x 2 4 2 1 09714 16 7 0 Var X X SD X 2 1 09714 1 04745 Let X be a continuous random variable with the cumulative distribution function F x 0 F x 3 x 0 x 0 x 2 8 F x 1 a 1 x2 x 2 Find the probability density function f x f x F x b x 0 f x F x 0 0 x 2 f x F x 3 8 x 2 f x F x 2 x3 Find the probability P 1 X 4 15 P 1 X 4 F 4 F 1 16 3 8 9 0 5625 16 OR 2 4 P 1 X 4 f x dx 2 x3 2 dx 0 5625 Find X E X E X X 2 x f x dx d dx 8 1 1 c 4 3 0 3 x dx 8 2 2 dx x3 x 1 3 x 2 2 7 3 2 1 75 1 0 2 8 2 4 4 x Find X SD X Var X X2 2 x f x dx However x 2 2 f x dx x 0 X 2 2 3 dx 8 x 2 Therefore Var X and SD X are not finite 2 2 x3 dx diverges
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