Chapter 2 Flashcards
Discrete uniform PMF
f(x) = P(x=x) = 1/b-a+1
Discrete uniform mean
E(x) = (a+b)/2
Discrete uniform VAR
VAR(x) = (b-a+1)^2-1/12
Binomial PMF
f(x) = P(x=x) = nCx · p^x · (1-p)^(n-x)
Mean binomial
E(x) = n · p
Binomial VAR
VAR(x) = n · p · (1-p)
Special property binomial
Sum of independents binomials with the same p equals B(n_i, p)
Hypergeometric PMF
f(x) = P(x=x) = (mCx) · (N-mCn-x)/(Ncn)
Mean hypergeo
E(x) = n · (m/N)
Var hypergeo
VAR(x) = ((n · m)/N) · ((N-m)/N) · (N-n/N-1)
PMF geometric
f(x) = P(x = x) = (1-p)^x-1
E(x) geo
E(x) = 1/p
geometric VAR(x) =
1-p/p^2
PMF geo fails
f(x)=P(x = x) = (1-p)^y · p
Mean geo fails
E(y) = (1/p)-1
Var geo fails
(1-p)/(p^2)
Special property geometric
(x-c|x > c)~X
(y-c|y e c)~Y
Negative binomial PMF
f(x) = P(x = x) = (x-1)C(r-1) · p^r · (1-p)^x-r
Mean negative binomial
r/p
VAR negative binomial
r · ((1-p/p^2)
Negative binomial fails PMF
f(x) = P(x = x) = (y+r-1)C(r-1) · p^r · (1-p)^y
Negative binomial fails mean
(r/p) - r
Negative binomial fails VAR
r((1-p)/p^2)
Special property negative binomial
Somme des binomiales négatives indépendantes X~N.B. (r_i, p)
PMF Poisson
f(x) = P(x = x) = (e^- λ) · (λ^x) / x!
E(x) Poisson
λ
VAR(x) Poisson
λ
Special property Poisson
Sum of λ
Uniform continuous PMF
f(x) = P(x = x) = 1/b-a
Uniform continuous CDF
F(x) = P(x ≤ x) = (x-a)/(b-a)
Uniform continuous mean
E(x) = a+b/2
Uniform continuous VAR
VAR(x) = (b-a)^2/12
Special property uniform continuous
(X | c < X < D) ~ Uniform(C, D)
(X - C|X > C) ~ Uniform(0, B - C)
Exponential PMF
f(x) = P(x = x) = 1/θ · e^(-x/θ)
Exponential CDF
F(x) = P(x ≤ x) = 1-e^(-x/θ)
Exponential mean
E(x) = θ
Exponential VAR
VAR(x) = θ^2
Special property Exponential
Memoryless property: (X C|X > C) ~ θ
PMF gamma
f(x) = P(x = x) = x^(a-1)/Γ(a) · θ^a · e^(-x/θ)
CDF gamma
1 − somme de P(Y = k) ,
Y ~ Poisson(𝜆 = x/θ)
𝛼 = 1, 2, 3, …
E(x) gamma
aθ
VAR(x) gamma
aθ^2
Special property Gamma
Sum of independent exponentials is a gamma (a,θ)
PMF CDF normal
Calculate Z and then follow the table of normal law.
E(x) normal
E(x) = µ
VAR(x) normal
VAR(x) = σ^2
Special property Normal
Sum of µ and of σ^2 creates a new normal law
PMF CDF lognormal
z = (ln x - µ)/σ then normal table
E(x) lognormal
E(x) = e^(µ+1/2σ^2)
VAR(x) lognormal
VAR(x) = e^(2µ+σ^2) · (e^(σ^2) - 1)
PMF beta
(Γ(a + b)/Γ(a)Γ(b)) · x^(a-1) · (1 - x)^(b-1)
E(x) beta
a/a+b
VAR(x) beta
ab/((a+b)^2 · (a+b+1))
Special property beta
B(1,1)~U(0,1)
F(x) = P(x ≤ x) = ?
Integral of f_x(s) evaluated between infiniti and x
S(x) = P(x > x) = ?
1 - F(x)
How do you calculate the mode?
f’(x) = 0 = mode, but in an exam you can trial and error the choices of the answer provided in the question. Substitute those choices by x and then take the biggest answer of them all.
Skewness ?
E((x-µ^3)/σ^3)
Kurtosis ?
E((x-µ^4)/σ^4)
First raw moment ?
mean E(x)
Second raw moment
E(x^2)
Second central moment
variance VAR(x)
VAR(x) = ?
E(x^2) - (E(x))^2
CV(x) = ?
SD(x)/E(x)
E(x+y)
E(x) + E(y)
VAR(x+y) dependant
VAR(x) + 2 Cov(x,y) + VAR(y)
VAR(x+y) independant
VAR(x) + VAR(y)
