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Probability (G11PRB)
Probability flash cards
56
Mathematics
Undergraduate 1
01/21/2012
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Term
Axiom 1
 Definition
P(E) ≥ 0 for any event E
Term
Axiom 2
 Definition
P(Ω) = 1
Term
Axiom 3
 Definition
if E1, E2, ... are such that Ei ∩ Ej = the empty set for i≠j then P(E1uE2u...) = P(E1) + P(E2) + ...
Term
P(E U F)
 Definition
P(E) + P(F) - P(E ∩ F)
Term
Multiplication rule
 Definition
Operation with k distinct stages and stage r has Nr possible outcomes, total outcomes = N1 x N2 x ... x Nk
Term
Permutation
 Definition
ordered arrangement of r objects
nPr = n!/(n-r)!
Term
Combination
 Definition
unordered arrangement of r objects
nCr = n!/(n-r)!r!
Term
P(E|F)
 Definition
P(E ∩ F) / P(F)
Term
Theorem of Total Probability
 Definition
E1, E2, ... partition of Ω
F is a subset of Ω
P(F) = Σi P(F|Ei)P(Ei)
Term
Bayes Theorem
 Definition
P(E|F) = P(F|E)P(E)/P(F)
Term
Probability mass function
 Definition
pX(x) = P(X=x)
Term
Cumulative distribution function
 Definition
FX(x) = P(X ≤ x)
Term
Mode of a random variable
 Definition
outcome with the highest probability
Term
Expectation
 Definition
Mean of X
∑x xpX(x)
Term
Variance
 Definition
measure of spread
E(X^2) - E(X)^2
Term
Independent events X + Y
 Definition
E(X+Y) = E(X) + E(Y)
Var(X+Y) = Var(X-Y) = Var(X) + Var(Y)
Term
Bernoulli distribution
 Definition
2 outcomes, success or failure
success occurs with probability p
failure with probability 1-p
E(X) = p, Var(X) = p(1-p)
Term
Binomial distribution: definition
 Definition
n independent Bernoulli trails
X~Bin(n,p)
Term
Binomial distribution: mass function
 Definition
pX(k) = (nCk) p^k (1-p)^(n-k)
Term
Binomial distribution: expectation and variance
 Definition
E(X) = np
Var(X) = np(1-p)
Term
Geometric distribution: definition
 Definition
X is the number of Bernoulli trials until success
X~Geom(p)
Term
Geometric: mass function
 Definition
pX(k) = (1-p)^(k-1) p
Term
Geometric: Expectation and variance
 Definition
E(X) = 1/p
Var(X) = (1-p)/p^2
Term
Poisson distribution: definition
 Definition
random events occurring at rate λ
X~Po(λ)
Term
Poisson: mass function
 Definition
pX(k) = e^-λ λ^k / k!
Term
Poisson expectation and variance
 Definition
E(X) = Var(X) = λ
Term
Negative binomial definition
 Definition
X is the number of trials until r successes
Term
negative binomial mass function
 Definition
pX(k) = (k-1)C(r-1) p^r (1-p)^(k-r)
Term
negative binomial expectation and variance
 Definition
E(X) = r/p
Var(X) = (1-p)/p^2
Term
discrete uniform definition
 Definition
X is a number randomly selected between a and a+b
Term
discrete uniform mass function
 Definition
pX(k) = 1/(b+1)
Term
discrete uniform expectation and variance
 Definition
E(X) = a + b/2
Var (X) = b(b+2)/12
Term
hypergeometric definition
 Definition
X is the number of elements A selected from n ≤ A + B
Term
hypergeometric mass function
 Definition
pX(k) = (ACk)(BC(n-k))/((A+B)Cn)
Term
hypergeometric expectation and variance
 Definition
E(X) = An/(A+B)
Var(X) = ABn(A+B+n)/[(A+B)^2 (A+B-1)]
Term
Expectation for continuous distribution
 Definition
E(X) = ∫ x fX(x) dx
Term
Median for continuous
 Definition
FX(x) = 0.5
Term
Mode for continuous
 Definition
fX is maximised
Term
Continuous uniform mass function
 Definition
1/(b-a) for x between a and b inclusive
0 otherwise
Term
Continuous uniform density function
 Definition
0 for x less than a
(x-a)/(b-a) for x between a and b including a
1 for x greater than or equal to b
Term
Continuous uniform expectation and variance
 Definition
E(X) = (1/2)(a+b)
Var(X) = (1/12)(b-a)^2
Term
Exponential density function
 Definition
1 - e^(-λx) for x greater than 0 inclusive
0 otherwise
Term
exponential mass function
 Definition
λe^(-λx) for x greater than 0 inclusive
0 otherwise
Term
exponential expectation and variance
 Definition
E(X) = 1/λ
Var(X) = 1/λ^2
Term
Normal mass function
 Definition
[1/(√(2π)σ)] e^[-(x-μ)^2 /2σ^2]
Term
Normal notation
 Definition
X~N(μ,σ^2)
Term
Normal expectation and variance
 Definition
E(X) = μ
Var(X) = σ^2
Term
Standard normal
 Definition
Z~N(0,1)
Term
Covariance
 Definition
Cov(X,Y) = E(XY) - E(X)E(Y)
Term
Correlation coefficient
 Definition
ρ(X,Y) = Cov(X,Y)/√(Var(X)Var(Y))
Term
Covariance of two independent X,Y
 Definition
Cov(X,Y) = 0
Term
Conditional bivariate distribution
 Definition
pX|Y (X|Y) = pXY(x,y) / pY(y)
Term
X and Y are independent
 Definition
fXY(x,y) = fX(x) fY(y)
Term
Central limit theorem
 Definition
Sn ≈ N(nμ, nσ^2)
where Sn = X1+X2+...+Xn
where Xn are independent and identically distributed with mean μand variance σ^2
Term
The sample mean (CLT)
 Definition
Xbar = Sn/n Xbar ≈ N(μ, σ^2/n)
Term
Normal approximation of binomial
 Definition
X is binomial (sum of n iid bernoulli trials) where E(Xi) = p and Var(Xi) = p(1-p)
apply CLT with limits to n≥20, np≥5, n(1-p)≥5
X ≈ N(np, np(1-p))
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