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Formula Sheet for Exam 1 - x dx 2 =Var(X)= x − x 2 f(x...

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Formula Sheet for Exam 1 (will be provided in the exam) P (A U B) = P( A) + P ( B ) - P ( A B ) Conditional probability : P ( A B ) P (A | B) = P(B) If A and B are independent, then P (A | B) = P (A) P (A ∩ B) = P(A)*P(B) Law of Total Probability : If P (A i ) ≠ 0 for each A i , P(B)= P (B|A 1 ) P(A 1 ) +…+ P (B|A n ) P(A n ) Bayes’ Rule: ( \ ) ( ) ( \ ) ( \ ) ( ) k k k i i i P B A P A P A B P B A P A = ( \ ) ( ) ( \ ) ( \ ) ( ) ( \ ) ( ) c c P B A P A P A B P B A P A P B A P A = + Discrete Probability Distributions : ( ) ( ) x x E X xP X x μ = = = 2 2 2 2 ( ) ( ) ( ) ( ) x x x x x Var X x P X x x P X x σ μ μ = = - = = = - Continuous Probability Distributions : P( a ≤ X≤ b) = b a dx x f ) ( F(x) =P(X ≤ x) = - x dt t f ) ( E(X) = xf ( x ) dx 2 σ =Var(X)= (
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Unformatted text preview: ( x ) dx 2 =Var(X)= ( ∫ x − x ) 2 f (x) dx Distribution Mean Variance pdf cdf Bernoulli(p) p p(1-p) p(x)=p x (1-p) 1-x Binomial(n,p) np np(1-p) x n p x p x n x f-- = ) 1 ( ) ( Geom(p) 1 p 2 1 p p-p(x)=p(1-p) x-1 F(x) = 1- (1-p) x Poisson(λ) λ λ ! ) ( x x e x f λ-= Exponential (λ) 1 2 1 f(x)= x e-x e--1 Normal(μ,σ 2 ) μ σ 2 , 2 2 1 2 2 1 ) ( --= σ μ πσ x e x f -= )! !*( ! x n x n x n...
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