Statistics for Research I
Statistics for Research I STAT 537
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This 16 page Class Notes was uploaded by Kamren McLaughlin on Monday October 26, 2015. The Class Notes belongs to STAT 537 at University of Tennessee - Knoxville taught by James Schmidhammer in Fall. Since its upload, it has received 23 views. For similar materials see /class/229891/stat-537-university-of-tennessee-knoxville in Statistics at University of Tennessee - Knoxville.
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Date Created: 10/26/15
Continuous Random Variables M3 of a continuous random variable Consider an experiment having an in nite number of possible outcomes Assign some real number to each possible outcome and associate this number with some random variable say X Example Weighing jellybeans Imagine purchasing a bag of 100 jelly beans and weighing each bean on a scale that is very accurate and very precise X weight of jellybean Def 3 of the probability density function of a continuous random variable X Let X be a continuous random variable assuming all values in some range Rx of the real line Then the probability density function of X is that function fX such that Pa g X g b fxdx Properties of f x l fx20 for allx lt2 jfltxgtdx1 Def 3 Mean of a continuous random variable X H EX jxfltxgtdx Def 3 Variance of a continuous random variable X 52 VarX EX 2 Zoe u2fxdx Def 3 Standard Deviation of random variable X 62 Def 3 Skewness of a continuous random variable X H3 Ix u3fxdx Def 3 Kurtosis of a continuous random variable X H4 Jx u4fxdx Def 3 Cumulative Distribution Function of a continuous random variable X Fx PX g x fooow The Normal Distribution M3 The normal d 3 A continuous random variable X is said to have a normal distribution with parameters u and 62 00 lt u lt oojcs2 gt 0 if the probability density function of X is 1 1xiH2 3 2 e G wltxltw V2756 Notation X Nu62 fx Moments Mean EX u Variance VarX 62 H2 Skewness u3zO gtBI 330 62 H4 Kurtos1s BZ 3 gtBZ O 4 G Normal Distribution p Sc p Zc u c p 6 26 36 Def 3 The standard normal d LL20 621 1 71x2 fx2e2 ooltxltoo TE Notation X NO1 Property If XNn62 and ifz X then 2 NO 1 The Standard Normal Distribution Standard Normal Distribution Pzgt17504006 Standard Normal Distribution 95994 0400 2 z175 Pzgt2370089 Standard Normal Distribution 9911 0089 4 3 2 1 o 1 2 2237 3 Reproductive Property Let c1x2xn be 11 independent normally distributed random variables with X Nuxics Let y ZaiXl i1 Then y N Hy i where My Zaiuxi i1 Central Limit Th m LindbergLevy Let c1x2xn be 11 iid random variables with EltXigtultoo VarXi 62 0 lt62 ltoo Let 1 n X XZ n 11 so that Y 2 H C32 63 11 Let Y X Hf X H Ti TN Then Y gtNO1 as n gtoo Application Normal Approximation to Binomial D Let X1Xn be iid Bernoulli random Variables so that Y 2 Zn X1 Binomial n p H with EY Hy quot9 VarY oi 71191 P Then Y Zz HyzY np gtNO1 as n gtoo 6y an 19 Rule of Thumb Normal Approximation to binomial is good if minnp nq 2 5 Example X Binomial n5 p04 3496 2592 2304 0778 0768 0102 3496 3 2592 2304 2 1 0778 0768 l 0102 For X Binomialn 519 04 M quotI 2 6inp1 p12 Using the exact binomial distribution Oquot quotOUOquot39U PX 1 00778 02592 2 03370 PXlt 1 Exact Probability 040 O 33 033 025 020 015 010 OCE Using normal approximation X np 1 2 PX 1P anG p 12 PZ 3 0913 01806 not so good PXlt 1 Normal Approxima on 040 O 33 033 OquotquotquotOUOquot39U
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