Probability Fundas

PROBABILITY FUNDAMENTALS PROF. NAVEEN BHATIA

PROBABILITY FUNDAMENTALS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAMENTALS ,[object Object],[object Object]

PROBABILITY FUNDAMENTALS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDMENTALS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDMENTALS ,[object Object],[object Object]

PROBABILITY FUNDMENTALS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDMENTALS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDMENTALS ,[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAMENTAL ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAMENTAL ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAMENTAL ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAMENTAL ,[object Object],SUB SYSTEM1 SUB SYSTEM2 SYSTEM

PROBABILITY FUNDAMENTALS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAMENTALS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAMENTALS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAMENTALS ,[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAMENTALS ,[object Object],[object Object],[object Object]

PROBABILITY FUNDAMENTALS 0 1 2 f(x) x

PROBABILITY FUNDAS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAS ,[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAS POSITIVE SKEW NEGATIVE SKEW

PROBABILITY FUNDAS ,[object Object],[object Object]

PROBABILITY FUNDAS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAS ,[object Object],Rx FFF 0 FFS FSF SFF FSS SFS SSF SSS 1 2 3 px q.q.q=q 3 3.p.q 2 3.p 2 q p 3

PROBABILITY FUNDAS ,[object Object],[object Object],[object Object]

PROBABILITY FUNDAS f(x) x λ EXPONENTIAL DENSITY FUNCTION

PROPBABILITY FUNDAS 1 F(X) x DISTRIBUTION FUNCTION 1-e - λ x

PROBABILITY FUNDAS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAS ,[object Object],f(x) x µ

BINOMIAL & B&S CONVERENCE ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

BINOMIAL & B&S CONVERENCE ,[object Object],[object Object],[object Object],[object Object],[object Object]

BINOMIAL & B&S CONVERENCE ,[object Object],θ IS THE TRUE PROBABILITY ASSOCIATED WITH INCREASE INS STOCK PRICE d= EXP(µτ/n -σ√τ/√n√(1-θ)/θ U= EXP(µτ/n +σ√τ/√n√(1-θ)/θ IF WE USE

BINOMIAL & B&S CONVERGENCE ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

LOGNORMAL DISTRIBUTION ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

LOGNORMAL DISTRIBUTION ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

LOGNORMAL DISTRIBUTION ,[object Object],[object Object],[object Object],[object Object]

LOGNORMAL DISTRIBUTION ,[object Object],[object Object],0

LOGNORMAL DISTRIBUTION ,[object Object],[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAS ,[object Object],α /2 α /2

PROBABILITY FUNDAS ,[object Object],[object Object],[object Object],[object Object]

PROBABILITY FUNDAS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],α /2 α /2 Χ 2 n-1 DISTRIBUTION 0 Χ 2 1- α /2,n-1 Χ 2 α /2,n-1

PROBABILITY FUNDAS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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