Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Probability exam, actuary SOA/CAS > Chapter 2 > Flashcards

Chapter 2 Flashcards

1
Q

Discrete uniform PMF

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Discrete uniform mean

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Discrete uniform VAR

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Binomial PMF

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

Mean binomial

A

E(x) = n · p

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

Binomial VAR

A

VAR(x) = n · p · (1-p)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Special property binomial

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

Hypergeometric PMF

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Mean hypergeo

A

E(x) = n · (m/N)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Var hypergeo

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

PMF geometric

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

E(x) geo

A

E(x) = 1/p

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

geometric VAR(x) =

A

1-p/p^2

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

PMF geo fails

A

f(x)=P(x = x) = (1-p)^y · p

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

Mean geo fails

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

Var geo fails

A

(1-p)/(p^2)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

Special property geometric

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

Negative binomial PMF

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

Mean negative binomial

A

r/p

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

VAR negative binomial

A

r · ((1-p/p^2)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

Negative binomial fails PMF

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

Negative binomial fails mean

A

(r/p) - r

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

Negative binomial fails VAR

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

Special property negative binomial

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

PMF Poisson

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
26
Q

E(x) Poisson

A

λ

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
27
Q

VAR(x) Poisson

A

λ

28
Q

Special property Poisson

A

Sum of λ

29
Q

Uniform continuous PMF

A

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

30
Q

Uniform continuous CDF

A

F(x) = P(x ≤ x) = (x-a)/(b-a)

31
Q

Uniform continuous mean

A

E(x) = a+b/2

32
Q

Uniform continuous VAR

A

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

33
Q

Special property uniform continuous

A

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

34
Q

Exponential PMF

A

f(x) = P(x = x) = 1/θ · e^(-x/θ)

35
Q

Exponential CDF

A

F(x) = P(x ≤ x) = 1-e^(-x/θ)

36
Q

Exponential mean

A

E(x) = θ

37
Q

Exponential VAR

A

VAR(x) = θ^2

38
Q

Special property Exponential

A

Memoryless property: (X C|X > C) ~ θ

39
Q

PMF gamma

A

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

40
Q

CDF gamma

A

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

41
Q

E(x) gamma

A

aθ

42
Q

VAR(x) gamma

A

aθ^2

43
Q

Special property Gamma

A

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

44
Q

PMF CDF normal

A

Calculate Z and then follow the table of normal law.

45
Q

E(x) normal

A

E(x) = µ

46
Q

VAR(x) normal

A

VAR(x) = σ^2

47
Q

Special property Normal

A

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

48
Q

PMF CDF lognormal

A

z = (ln x - µ)/σ then normal table

49
Q

E(x) lognormal

A

E(x) = e^(µ+1/2σ^2)

50
Q

VAR(x) lognormal

A

VAR(x) = e^(2µ+σ^2) · (e^(σ^2) - 1)

51
Q

PMF beta

A

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

52
Q

E(x) beta

A

a/a+b

53
Q

VAR(x) beta

A

ab/((a+b)^2 · (a+b+1))

54
Q

Special property beta

A

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

55
Q

F(x) = P(x ≤ x) = ?

A

Integral of f_x(s) evaluated between infiniti and x

56
Q

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

A

1 - F(x)

57
Q

How do you calculate the mode?

A

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.

58
Q

Skewness ?

A

E((x-µ^3)/σ^3)

59
Q

Kurtosis ?

A

E((x-µ^4)/σ^4)

60
Q

First raw moment ?

A

mean E(x)

61
Q

Second raw moment

A

E(x^2)

62
Q

Second central moment

A

variance VAR(x)

63
Q

VAR(x) = ?

A

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

64
Q

CV(x) = ?

A

SD(x)/E(x)

65
Q

E(x+y)

A

E(x) + E(y)

66
Q

VAR(x+y) dependant

A

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

67
Q

VAR(x+y) independant

A

VAR(x) + VAR(y)

Probability exam, actuary SOA/CAS (4 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions