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# 479 Class Note for STAT 416 at PSU

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This 9 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15

Random Variables 0 Functions of random outcomes 1 Flip a cointwice H H H T T H T Let the number of heads in the two ips be X TTX 0 HTTHX1 HHX 2 0 Random variables realvalued functions de ned on the sample space Discrete The RV takes value on either a nite or countable number of possible values Example Toss a coin until the rst head appears Let N be the number oftosses N 1 2 Continuous The RV takes a continuum of possi ble values Example The lifetime of a car Cumulative Distribution Functions 0 Let X be a RV The cumulative distribution function cd ofX is Fb PX g b Toss a coin until the rst head appears N is the number of tosses PN g b PN1PN2PN W pp1 pp1 pLbJ1 1 1 29W 0 Properties of Fb 1 Fb is nondecreasing function of b 2 hrnbOO Fb Foo 1 3 limboo Fb F oo 0 4 Fb is rightcontinuous Discrete Random Variables o X take at most countably many possible values 0 Probability massfunctl39on pa pa PX 0 Let the possible values be 51 239 1 2 2 0 Mr 0 for all other x 39 1 0 Important discrete random variables 1 Bernoulli random variable 0 X PX1p PXO1 p 2 Binomial RV Flip an unfair coin n times Assume the experi ments are independent trials Let X be the number of heads in the n trials The range ofX 0 1 PXk Zpk1 pnkkO1n If de ne X as the indicator for whether the ith ip is a head X 1 for a head and X 0 for a tail X is a Bernoulli RV X iXZ39 11 3 Geometric RV An unfair coin has probability p of coming up heads Flip the coin until the rst head appears Let X be the number of ips needed X 12 PX n PThe rst n l ips are tails the nth ip is head 1 mm 4 Poisson RV X 0 12 Continuous Random Variables o X takes on a continuum of possible values 0 Probability density function pd9 f PX e B mm B 0 Properties 1 f 2 0 2 ff fxdx 1 0 Calculate probability b pm g X g b fcdr 0 Relationship with F a and fa Fm PX e ooa L mm dF a da fa 0 Important continous RVs 1 Uniform 1 O lt I lt1 x 7 0 otherwise More generally fltxlb a altxltb 0 otherwise 2 Exponential for any gt 0 Ae 1 Z 0 f 0 otherwise 3 Gamma RV for any gt 0 o gt 0 A67Axail x Z 0 0 otherwise 4 Normal RV for any 0 gt 0 and any u l 2 2 i 95 M 20 1 7 e 00 lt x lt 00 f Tm XNNw l If X N N M 02 then for any constants a and b aX b N Nau b 202 Expectation 0 Discrete case EX Zwmw l Bernoulli EX p 2 Binomial EX np 3 Geometric EX 1 4 Poisson with parameter oz EX oz 0 Continuous case EX ff xf 1 Uniform on a b EX 2 Exponential with parameter A EX l 3 Gamma with A oz EX oz 4 Gaussian NM 02 EX Lz o Expectaion of a function of a RV 0 Proposition 21 a If X is a discrete RV with pmf Mm then for any realvalued function g E95II Z 956p56 39p gt0 b If X is a continuous RV with pdf f x for any realvalued function g 0 Corollary 22 If a and b are constants then E aX b aE X b Variance 0 De nition VarX EX VarX EX2 EX2 0 Discrete l Bernoulli VarX pl p 2 Binomial VarX npl p 3 Geometric VarX 4 Poisson VarX or same as EX 0 Continuous 1 Uniform on 01 b VarX b12602 2 Exponential VarX 3 Gaussian VarX 02 4 Gamma VarX oz2

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