STAT 400 Lecture AL1 Fall 2014 Dalpiaz Discrete vs Continuous RVs random variables discrete continuous probability mass function p m f probability density function p d f p x P X x f x x x 0 p x 1 f x 0 p x 1 f x d x all x 1 cumulative distribution function c d f F x P X x F x x p y F x y x f y d y expected value E X X discrete E X x p x continuous E X all x discrete E g X g x p x all x x f x d x continuous E g X g x f x d x variance Var X 2X E X X 2 E X 2 E X 2 discrete Var X continuous x X p x 2 x X Var X all x 2 f x dx x 2 p x x 2 f x dx E X 2 E X 2 all x moment generating function MX t E e t X discrete MX t continuous e t x p x MX t all x Example 1 Let X be a continuous random variable with the probability density function a f x k x 2 0 x 1 f x 0 otherwise What must the value of k be so that f x is a probability density function 1 f x 0 e 2 f x dx 1 t x f x dx f x dx 1 1 k x b dx k x 2 dx 0 x3 1 0 3 k 1 2 0 1 k 3 k 3 k 3 Find the cumulative distribution function F x P X x 2 f X x 3 x 0 x 1 0 x 0 F X x 0 0 x 1 FX x o w x 3y 0 x 1 fX x c FX x Find the probability P 0 4 X 0 8 P 0 4 X 0 8 0 8 f x dx 0 4 0 8 3 x 0 4 2 dx x 3 0 8 0 4 0 8 3 0 4 3 0 448 F X x 1 2 d y x 3 OR P 0 4 X 0 8 F X 0 8 F X 0 4 0 8 3 0 4 3 0 448 d Find the median of the distribution of X m Need m such that Area to the left of m f x dx 1 2 m e m f x dx 3 1 m 3 x 2 dx x 3 0 m 0 1 2 m3 0 7937 2 Find X E X E X X x f x dx x 3 x 1 dx 3 x 3 dx 1 2 0 0 x4 1 3 0 75 3 4 0 4 f Find X SD X Var X X 2 x 2 f x dx 2 X 2 1 3 x 4 dx 3 4 0 x 5 1 3 2 3 9 3 0 0375 3 5 0 4 5 16 80 X SD X Var X 0 0375 0 19365 g Find the moment generating function of X M X t MX t E e t X e t x f x dx dv e t x dx du 6 x dx v MX t 0 t x 3 x 2 dx 1 t e t x 1 1 e t x 3 x 2 dx 3 x 2 e t x t 0 3 et t 1 1 0 1 t x e 6 x dx t 1 t x 6 x dx t e 0 u 6 x dv du 6 dx v 1 t 3 MX t e t t e 0 u 3 x 2 1 1 2 1 t e t x dx e t x 1 1 1 3 t 1 1 t x t x e e 6 x dx e 6 x 2 t t t 0 0 0 1 t 2 e t x 6 dx 6 1 3 t 6 3 6 6 6 e 2 e t 3 e t x e t 2 e t 3 e t 3 t t t t t t t 0 t 0 M X 0 1 h Find E X and E ln X 1 E X 0 x 3x 2 d x 6 7 1 E ln X ln x 3 x 0 2 1 dx 3 Example 2 6 f X x 5 x x 1 0 x 1 F X x 0 x 1 FX x o w x 5y 6 dy 1 y 5 fX x x 1 1 x 5 FX x 1 1 5 6 x 5x d x 5x d x E X X 2 E X x 2 5x 6 1 Var X 5 1 25 4 5 d x 5 x 4 d x 3 1 E X 2 E X 2 E X 10 does NOT exist since x 10 5 5 3 4 2 5 48 5 x 6 d x diverges 1 Median FX m 30th percentile 1 2 F X 0 30 0 30 1 m 5 1 2 m 5 2 1 1487 1 0 30 5 0 30 0 30 5 1 1 07394 0 70 Example 3 Suppose a random variable X has the following probability density function f x C e a x 0 x 1 0 otherwise What must the value of C be so that f x is a probability density function For f x to be a probability density function we must have f x 0 1 f x dx 1 2 1 1 f x dx C e C e x 10 C e Therefore b 1 C e x dx dx 0 x 0 C 1 e 1 e 1 e C 1 5819767 e 1 e x e 1 e f x 0 0 x 1 otherwise Find X E X X E X 1 0 x f x dx x e e x dx e 1 1 e x x e dx e 1 0 Integrating by parts 1 x e x dx x e x 0 e 1 1 x e dx 0 1 0 e 1 x dx 0 e 1 e x 10 1 2 e 1 e 2 e Therefore e 1 x e x dx e e 2 e 2 0 418 X E X e 1 0 c e 1 e e 1 Find the cumulative distribution function F x P X x x F x P X x F x 0 for x 0 f y dy F x 1 for x 1 For 0 x 1 F x x f y dy x e y e 1 e dy 0 d e x 1 e e 1 x e e 1 e 1 e x 0 e e x 1 F x x e 1 e 1 Find the median of the probability distribution of X Need m such that P X m P X m 1 2 m e …
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