Chapter 2

Question 1

Discrete uniform PMF

Answer 1

f(x) = P(x=x) = 1/b-a+1

Question 2

Discrete uniform mean

Answer 2

E(x) = (a+b)/2

Question 3

Discrete uniform VAR

Answer 3

VAR(x) = (b-a+1)^2-1/12

Question 4

Binomial PMF

Answer 4

f(x) = P(x=x) = nCx · p^x · (1-p)^(n-x)

Question 5

Mean binomial

Answer 5

E(x) = n · p

Question 6

Binomial VAR

Answer 6

VAR(x) = n · p · (1-p)

Question 7

Special property binomial

Answer 7

Sum of independents binomials with the same p equals B(n_i, p)

Question 8

Hypergeometric PMF

Answer 8

f(x) = P(x=x) = (mCx) · (N-mCn-x)/(Ncn)

Question 9

Mean hypergeo

Answer 9

E(x) = n · (m/N)

Question 10

Var hypergeo

Answer 10

VAR(x) = ((n · m)/N) · ((N-m)/N) · (N-n/N-1)

Question 11

PMF geometric

Answer 11

f(x) = P(x = x) = (1-p)^x-1

Question 12

E(x) geo

Answer 12

E(x) = 1/p

Question 13

geometric VAR(x) =

Answer 13

1-p/p^2

Question 14

PMF geo fails

Answer 14

f(x)=P(x = x) = (1-p)^y · p

Question 15

Mean geo fails

Answer 15

E(y) = (1/p)-1

Question 16

Var geo fails

Answer 16

(1-p)/(p^2)

Question 17

Special property geometric

Answer 17

(x-c|x > c)~X
(y-c|y e c)~Y

Question 18

Negative binomial PMF

Answer 18

f(x) = P(x = x) = (x-1)C(r-1) · p^r · (1-p)^x-r

Question 19

Mean negative binomial

Answer 19

r/p

Question 20

VAR negative binomial

Answer 20

r · ((1-p/p^2)

Question 21

Negative binomial fails PMF

Answer 21

f(x) = P(x = x) = (y+r-1)C(r-1) · p^r · (1-p)^y

Question 22

Negative binomial fails mean

Answer 22

(r/p) - r

Question 23

Negative binomial fails VAR

Answer 23

r((1-p)/p^2)

Question 24

Special property negative binomial

Answer 24

Somme des binomiales négatives indépendantes X~N.B. (r_i, p)

Question 25

PMF Poisson

Answer 25

f(x) = P(x = x) = (e^- λ) · (λ^x) / x!

Question 26

E(x) Poisson

Answer 26

λ

Question 27

VAR(x) Poisson

Answer 27

λ

Question 28

Special property Poisson

Answer 28

Sum of λ

Question 29

Uniform continuous PMF

Answer 29

f(x) = P(x = x) = 1/b-a

Question 30

Uniform continuous CDF

Answer 30

F(x) = P(x ≤ x) = (x-a)/(b-a)

Question 31

Uniform continuous mean

Answer 31

E(x) = a+b/2

Question 32

Uniform continuous VAR

Answer 32

VAR(x) = (b-a)^2/12

Question 33

Special property uniform continuous

Answer 33

(X | c < X < D) ~ Uniform(C, D) (X - C|X > C) ~ Uniform(0, B - C)

Question 34

Exponential PMF

Answer 34

f(x) = P(x = x) = 1/θ · e^(-x/θ)

Question 35

Exponential CDF

Answer 35

F(x) = P(x ≤ x) = 1-e^(-x/θ)

Question 36

Exponential mean

Answer 36

E(x) = θ

Question 37

Exponential VAR

Answer 37

VAR(x) = θ^2

Question 38

Special property Exponential

Answer 38

Memoryless property: (X  C|X > C) ~ θ

Question 39

PMF gamma

Answer 39

f(x) = P(x = x) = x^(a-1)/Γ(a) · θ^a · e^(-x/θ)

Question 40

CDF gamma

Answer 40

1 − somme de P(Y = k) , Y ~ Poisson(𝜆 = x/θ) 𝛼 = 1, 2, 3, …

Question 41

E(x) gamma

Answer 41

aθ

Question 42

VAR(x) gamma

Answer 42

aθ^2

Question 43

Special property Gamma

Answer 43

Sum of independent exponentials is a gamma (a,θ)

Question 44

PMF CDF normal

Answer 44

Calculate Z and then follow the table of normal law.

Question 45

E(x) normal

Answer 45

E(x) = µ

Question 46

VAR(x) normal

Answer 46

VAR(x) = σ^2

Question 47

Special property Normal

Answer 47

Sum of µ and of σ^2 creates a new normal law

Question 48

PMF CDF lognormal

Answer 48

z = (ln x - µ)/σ then normal table

Question 49

E(x) lognormal

Answer 49

E(x) = e^(µ+1/2σ^2)

Question 50

VAR(x) lognormal

Answer 50

VAR(x) = e^(2µ+σ^2) · (e^(σ^2) - 1)

Question 51

PMF beta

Answer 51

(Γ(a + b)/Γ(a)Γ(b)) · x^(a-1) · (1 - x)^(b-1)

Question 52

E(x) beta

Answer 52

a/a+b

Question 53

VAR(x) beta

Answer 53

ab/((a+b)^2 · (a+b+1))

Question 54

Special property beta

Answer 54

B(1,1)~U(0,1)

Question 55

F(x) = P(x ≤ x) = ?

Answer 55

Integral of f_x(s) evaluated between infiniti and x

Question 56

S(x) = P(x > x) = ?

Answer 56

1 - F(x)

Question 57

How do you calculate the mode?

Answer 57

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.

Question 58

Skewness ?

Answer 58

E((x-µ^3)/σ^3)

Question 59

Kurtosis ?

Answer 59

E((x-µ^4)/σ^4)

Question 60

First raw moment ?

Answer 60

mean E(x)

Question 61

Second raw moment

Answer 61

E(x^2)

Question 62

Second central moment

Answer 62

variance VAR(x)

Question 63

VAR(x) = ?

Answer 63

E(x^2) - (E(x))^2

Question 64

CV(x) = ?

Answer 64

SD(x)/E(x)

Question 65

E(x+y)

Answer 65

E(x) + E(y)

Question 66

VAR(x+y) dependant

Answer 66

VAR(x) + 2 Cov(x,y) + VAR(y)

Question 67

VAR(x+y) independant

Answer 67

VAR(x) + VAR(y)