任務(wù)講義-v2frm強化定量分析

FRM一級培訓(xùn)項目tative

ysis講師：金程教育資深培訓(xùn)師地點：

■

□

□【夢軒考資

】6454842

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FRM全程+講義【夢軒考資

】tative

ysis1.

ProbabilityBasic

StatisticsDistributionsHypothesis

Tests

and

Confidence

IntervalsLinear

RegressionSimulation

ModelingEstimating

Volatilities

and

Correlations專業(yè)提供CFA

FRM全程+講義2-148If

A,B,C…are

mutually

exclusive

events,P

A

B

C

...

PA

PB

PC

...If

A,B,C,…are

mutually

exclusive

and

collectively

exhaustive

set

ofevents,PA

B

C

...

PA

PB

PC

...

1Generally,PA

B

PA

PB

PAB【夢軒考資

】ProbabilityProperties

of

Probabilities專業(yè)提供CFA

FRM全程+講義3-148Unconditional

probability:

P(A),

P(B)Conditional

probability:

P(A|B)【夢軒考資

】ProbabilityJoint

probability:

P(AB)

=

P(A)P(B|A)

=

P(B)P(A|B)PA|B

PAB

PBPAB

PA;

PB

0PB|A

; PA

0專業(yè)提供CFA

FRM全程+講義4-148Joint

probability:

P(AB)Multiplicationrule:P(AB)

=

P(A|B)×P(B)

=

P(B|A)×P(A)If

A

and

B

are

mutually

exclusive

events,

then:P(AB)

=

P(A|B)=

P(B|A)=

0Probability

that

at

least

one

of

two

events

will

occur:Addition

rule:P(A

or

B)

=

P(A)

+

P(B)

–

P(AB)If

A

and

B

are

mutually

exclusive

events,

then:P(A

or

B)

=

P(A)

+

P(B)【夢軒考資

】Probability專業(yè)提供CFA

FRM全程+講義5-148The

occurrence

of

A

has

no

influence

of

on

the

occurrence

of

BP(A|B)

=

P(A)

or

P(B|A)

=

P(B)P(AB)

=

P(A)×P(B)P(A

or

B)

=

P(A)

+

P(B)

–

P(AB)Independence

and

Mutually

Exclusive

are

quite

differentIf

exclusive,

must

not

independence;Cause

exclusive

means

if

A

occur,

B

can

not

occur,

A

influents

B.P(AB)

=

P(A)×P(B)【夢軒考資

】Probability專業(yè)提供CFA

FRM全程+講義6-148Probability

DistributionDescribe

the

probabilities

of

all

the

possiblevariable.Discrete

and

continuous

random

variablesDiscrete

random

variables:

the

number

of

possiblees

for

a

randomes

can

becounted,

and

for

each

possible e,

there

is

a

measurable

andpositive

probability.Continuous

variables:

the

number

of

possibleeven

if

lower

and

upper

bounds

exist.P(x)

=

0

even

though

x

can

occur.P

(x1<X<x2)es

is

infinite,【夢軒考資

】Probability專業(yè)提供CFA

FRM全程+講義7-148【夢軒考資

】ProbabilityProbability

function: p(x)

=

P(X=x)For

discrete

random

variables0

≤

p(x)

≤

1Σp(x)

=

1Probability

density

function

(p.d.f)

:

f(x)For

continuous

random

variable

commonlyCumulative

probability

function

(c.p.f)

:

F(x)Discrete:

(x)

=

P(X

≤x)Continuous:xf

uduF(x)

專業(yè)提供CFA

FRM全程+講義8-148ProbabilityF(x)1【夢軒考資

】Probabilityf(x)xx00baF(b)F(a)ba}P(a

X

b)

=

Area

under

f(x)between

a

and

b=

F(b)

-

F(a)P(a

X

b)=F(b)

-

F(a)專業(yè)提供CFA

FRM全程+講義9-148Properties

of

CDFF(-∞)

=

0

and

F(∞)

=

1F(X)

is

a

non-decreasing

function

such

that

if

x2

>

x1

then

F(x2)

≥

F(x1)P(X

≥

k)

=

1

–

F(k)P(x1

≤

X

≤

x2)

=

F(x2)

–

F(x1)Probability

MatrixSummarize

joint

probabilities

in

a

probability

matrix.Unconditional/marginal

probabilities

can

be

seen

by

adding

across

arow

or

down

a

column.【夢軒考資

】Probability專業(yè)提供CFA

FRM全程+講義10-148Consider

two

stocks.

Assume

that

both

Stock

S

and

Stock

T

caneach

only

reach

three

price

levels.

Stock

S

can

achieve:

$10,

$15,or

$20;

Stock

T

can

achieve:

$15,

$20,

or

$30.【夢軒考資

】ProbabilityExampleJoint

Probability: P(S=$20,

T=$30)

=

3/26Marginal/unconditional

probability: P(S

=

$20)

=

(2+3+3)/26

=

8/26Conditional

ProbabilityS=$10S=$15S=$20TotalT=$150224T=$2034310T=$3036312Total6128263

26

310

26

10PS

$20

T

$20

PS

$20,T

$20

PT

$20專業(yè)提供CFA

FRM全程+講義11-148Bayes’

TheoremExampleJohn

is

forecasting

a

stock’s

performance

in

2010

conditional

onthe

state

of

the

economy

of

the

country

in

which

the

firm

is

based.He

divides

the

economy’s

performance

into

three

categories

of“GOOD”,

“NEUTRAL”

and

“POOR”

and

the

stock’s

performanceinto

three

categories

of

“increase”,

“constant”

and

“decrease”.Estimate

the

probability

that

the

state

of

the

economy

is

NEUTRALgiven

that

the

stock

performance

is

constant.GoodNeutralPoorIncrease16%15%7.5%Constant2%6%7.5%Decrease2%9%35%38.5%15.5%46.0%20%

30%

50%

100%Pneutral

constant6%

38.71%15.5%PBPAB

PB

APA

PB

APA

PB

APA

PB

APA【夢軒考資

】Probability專業(yè)提供CFA

FRM全程+講義12-148ExerciseThe

following

is

a

probability

matrix

for

X

=

{1,

2,

3}

and

Y

=

{1,

2,

3}XYEach

of

the

following

is

true

except:X

and

Y

are

independentThe

Covariance

(X,Y)

is

non-zeroThe

probability

Y

=

3

conditional

on

X

=

1

is

10%The

unconditional

probability

that

X

=

2

is

50%.12316%15%9%212%30%18%32%5%3%【夢軒考資

】

6454842

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FRM全程

+講義13-148ExerciseNext

year

the

economy

will

experience

one

of

three

states:

downturn,stable

state,

or

growth.

The

following

probability

matrix

is

as

follow:EconomyBondIf

we

observe

that

the

bond

has

defaulted,

what

is

the

probability

that

theeconomy

experienced

a

downturn?0.60%19.40%26.33%31.58%DownturnStableGrowthSurvive19.40%49.00%29.70%Default0.60%1.00%0.30%【夢軒考資

】

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FRM全程

+講義14-148ExerciseThere

is

a

unconditional

probability

of

20%

that

the

Fed

will

initiate

QE4.If

the

Fed

announces

QE4,

then

ABC

hedge

fund

will

outperform

themarket

with

a

70%

probability.

If

the

fed

does

not

announce

QE

4,

thereis

only

a

40%

probability

that

ABC

will

outperform.

If

we

observe

thatABC

outperforms

the

market,

which

is

nearest

to

the

probability

that

theFed

announced

QE4?A.

20%B.28%C.30%D.42%【夢軒考資

】

6454842

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FRM全程

+講義15-148概率條件概率和非條件概率（邊際概率）邊際概率與聯(lián)合概率之間的關(guān)系？概率矩陣的運用及

公式的計算？【夢軒考資

】6454842

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FRM全程+講義16-148【夢軒考資

】tative

ysis1.

Probability2.

Basic

StatisticsDistributionsHypothesis

Tests

and

Confidence

IntervalsLinear

RegressionSimulation

ModelingEstimating

Volatilities

and

Correlations專業(yè)提供CFA

FRM全程+講義17-148【夢軒考資

】Expected

ValueExpected

Value:

A

Measure

of Central

TendencyProperties

of

Expected

Value1.

If

b

is

a

constant,

E(b)

=

b2.

E(X+Y)

=

E(X)

+

E(Y)3.

In

general,

E(XY)

≠

E(X)E(Y);

If

X

and

Y

are

independent

randomvariables,

then

E(XY)

=

E(X)E(Y)4.

E(X2)

≠

[E(X)]25.

If

a

is

a

constant,

E(aX)

=

aE(X)6.

If

a

and

b

are

constants,

then

E(aX+b)

=

aE(X)

+

E(b)

=

aE(X)

+

b專業(yè)提供CFA

FRM全程+講義18-148VariancexThe

positive

square

root

of ,

σ is

known

as

the

standard

deviation.2xAbove

formula

is

the

definition

of

variance.

To

compute

the

variance,we

use

the

following

formula:var

X

EX2

EX2

Variance:

a

Measure

of

Dispersion

–

the

second

momentThe

definition

of

variancek

22XXix

ii1222X2XE

X

E

X

2

variance

X

E

X

X

P

E

X

【夢軒考資

】6454842

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FRM全程+講義19-148VarianceProperties

of

Variance1.

The

variance

of

a

constant

is

zero.

Bydefinition,a

constant

has

novariability.2.

If

X

and

Y

are

two

independent

random

variables,

thenvar(X+Y)

=

var(X)

+

var(Y)

and

var(X

–

Y)

=

var(X)

+

var(Y)3.

If

b

is

a

constant,

then:

var(X

+

b)

=

var(X)4.

If

a

is

constant,

then:

var(aX)

=

a2var(X)5.

If

a

and

b

are

constant,

then:

var(aX+b)

=

a2var(X)6.

If

X

and

Y

are

independent

random

variables

and

a

and

b

areconstants,

then

var(aX

+

bY)

=

a2var(X)

+

b2var(Y)7.

For

computational

convenience,

we

can

get:var(X)

=

E(X2)

-

[E(X)]2

,

thatEX2

x2

pxx【夢軒考資

】6454842

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FRM全程+講義20-148【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程Sample

Mean

and

VarianceSample

MeanThe

sample

mean

of

a

r.v.

X

is

generally

denoted

by

the

symbolThe

sample

mean

is

known

as

an

estimator

of

E(X),

which

we

can

now

callthe

population

mean.An

estimate

of

the

population

is

simply

the

numerical

value

taken

by

anestimator.Sample

Variance,

which

wecan

now

call

the

population

variance.

The

sample

variance

is

defined

as

:If

the

sample

size

is

reasonably

large,

wecan

divide

by

n

instead

of

(n-1).The

expression

(n-1)

is

known

as

the

degrees

of

freedom.XX

and

isndefined

as

:

X

Xini1xS

(

the

positive

square

root

of

S2

),

is

called

the

sample

standarddeviation.2x

2xS2ii1X

X1n

1nxThe

sample

variance,

denoted

by

S2

which

is

an

estimator

of+講義21-148CovarianceCovariance

measures

how

one

random

variable

moves

with

anotherrandom

variable.Covariance

ranges

from

negative

infinity

to

positive

infinity.Properties

of

CovarianceIf

X

and

Y

are

independent

random

variables,

their

covariance

is

zero.If

X

and

Y

are

not

independent,

then:Cov

a

bX,

c

dY

b

d

Cov

X,YVar

X

Y

Var

X

Var

Y

2Cov

X,

YCov

X,

X

EX-

EX

X

-

EX

2

X

E

X

X

Y

Y

EXY

EXEY【夢軒考資

】CovarianceXYsXYi

iX

X

Y

Y

n

1

i11n專業(yè)提供CFA

FRM全程+講義22-148【夢軒考資

】Correlation

coefficientCorrelation

coefficientProperties

of

Correlation

coefficientCorrelation

measures

the

linear

relationship

between

two

randomvariables.Correlation

has

no

units,

ranges

from

–1

to

+1.If

two

variables

are

independent,

their

covariance

is

zero,

therefore,the

correlation

coefficient

will

be

zero.

The

converse,

however,

is

nottrue.

For

example,

Y

=

X2Variances

of

correlated

Variables.var

X

Y

var

X

var

Y

2x

yXYXYX

Y,

cov

X,Y

E

X

Y

X

YXYXYX

YsXYr

s

s

6454842

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FRM全程+講義23-148【夢軒考資

】SkewnessSkewnessA

measure

of

asymmetry

of

a

PDFPositive

skewed：Mode<median<mean,

having

a

right

fat

tailNegative

skewed：Mode>media>mean,

having

a

left

fat

tailPositive-SkewedMode

Median

MeanNegative-SkewedMean

Median

ModeSymmetricMean

=

Median

=

Mode

3xxSymmetrical

and

nonsymmetrical

distributionsPositively

skewed

(right

skewed)

and

negatively

skewed(left

skewed)E

X

3S

專業(yè)提供CFA

FRM全程+講義24-148KurtosisKurtosisA

measure

of

tallness

or

flatness

of

a

PDFExcess

kurtosis

=

kurtosis

-

3

4x22xE

X

K

E

X

LeptokurticMesokurticPlatykurticKurtosis>3=3<3Excess

kurtosis>0=0<0Tails

(assuming

samevariation)Fat

tailnormalThin

tail【夢軒考資

】

6454842

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FRM全程+講義25-148LeptokurticNormalDistributionFat

tailKurtosis【夢軒考資

】6454842

專業(yè)提供CFA

FRM全程+講義26-148Provides

a

shorthand

method

for

specifying

a

cumulative

probabilitywithout

our

need

to

know

the

underlying

distribution

(conditional

on

afinite

variance).P(|X-μ|

≤

kσ)

≥

1

–

1/k2，k>1【夢軒考資

】Chebyshev’s

InequalityChebyshev’s

Inequality1

1221

14

3

475%1

1321

1

98

989%1

11

1

1594%421616AtleastLiewithinStandard

deviationsof

the

mean2346454842

專業(yè)提供CFA

FRM全程+講義27-148ExerciseA

model

of

the

frequency

of

losses

(L)

per

day

assumes

the

followingdiscrete

distribution:

zero

loss

with

probability

of

20%;

one

loss

withprobability

of

30%;

two

losses

with

probability

of

30%;

three

losses

withprobability

of

10%;

and

four

losses

with

probability

of

10%.

What

are,respectively,

the

expected

number

of

loss

events

and

the

standarddeviation

of

the

number

of

loss

events?E(L)

=

1.2

and

σ

=

1.44E(L)

=

1.6

and

σ

=

1.20E(L)

=

1.8

and

σ

=

2.33E(L)

=

2.2

and

σ

=

9.60【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程

+講義28-148ExerciseAn

yst

is

concerned

with

the

symmetry

and

peakedness

of

adistribution

of

returns

over

a

period

of

time

for

a

company

she

isexamining.

She

does

some

calculations

and

finds

that

the

median

returnis

4.2%,

the

mean

return

is

3.7%,

and

the

mode

return

is

4.8%.

She

alsofinds

that

the

measure

of

kurtosis

is

2.

Based

on

this

information,

thecorrect

characterization

of

the

distribution

of

return

over

time

is:SkewnessPositivePositiveKurtosisLeptokurticPlatykurticC.

Negative

PlatykurticD.

Negative

Leptokurtic【夢軒考資

】6454842

專業(yè)提供CFA

FRM全程+講義29-148ExerciseUsing

Chebyshev’s

inequality,

what

is

the

proportion

of

observationsfrom

a

population

of

250

that

must

lie

within

three

standard

deviations

ofthe

mean,

regardless

of

the

shape

of

the

distribution?A.

75%B.99%C.

89%D.

54%【夢軒考資

】6454842

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FRM全程+講義30-148【夢軒考資

】統(tǒng)計學(xué)基礎(chǔ)均值、方差、協(xié)方差、相關(guān)系數(shù)計算？組合方差/波動率計算？偏度的基本性質(zhì)及分類？峰度的基本性質(zhì)及分類？契比雪夫不等式計算？專業(yè)提供CFA

FRM全程+講義31-148【夢軒考資

】tative

ysisProbabilityBasic

Statistics3.

DistributionsHypothesis

Tests

and

Confidence

IntervalsLinear

RegressionSimulation

ModelingEstimating

Volatilities

and

Correlations專業(yè)提供CFA

FRM全程+講義32-148Discrete

Probability

DistributionBinomial

DistributionBernoulli

Random

VariableP(Y

=

1)

=

pP(Y

=

0)

=

1

–

pBinomial

random

variable

the

probability

of

x

successes

in

n

trailsExpectations

and

variances

x

xnnxp

x

P X

x

C

p 1

ppx

1

pnxn!x!n

x!ExpectationVarianceBernoulli

random

variable

(Y)pp(1

–

p)Binomial

random

variable

(X)npnp(1

–

p)【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程+講義33-148Discrete

Probability

DistributionPoisson

DistributionWhen

there

are

a

large

number

of

trials

but

a

small

probability

ofsuccess,

binomial

calculations e

impractical.If

we

substitute

λ/n

for

p,

and

let

n

very

large,

the

binomial

distributionλ

indicates

the

rate

of

occurrence

of

the

random

events;

i.e.,

it

lsus

how

many

events

occur

o age

per

unit

of

time.

es

thepPokissoPndXistribkution.ke

,

npk!【夢軒考資

】

6454842

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FRM全程+講義34-148【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程Continuous

Probability

DistributionContinuous

Uniform

DistributionProbability

density

functionCumulative

distribution

function1a

x

botherwise,,f

x

b

a

0Fx

x

a

,b

a0

,

for x

afor a

x

b

1, for

x

bab+講義35-148Some

Important

Probability

DistributionsProperties

of

Continuous

Uniform

DistributionE(X)

=

(a

+

b)/2Var(X)

=

(b

–

a)2/12For

all

a

≤

x1

<

x2

≤

b,

we

have:x

2Px1

X

x2

f

xdx

x2

x1

/

b

ax1【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程

+講義36-148Continuous

Probability

DistributionNormal

DistributionX~N(μ,

σ2),

fully

described

by

its

two

parameters

μ

and

σ2.Bell-shaped,

symmetrical

distribution:

skewness

=

0,

kurtosis

=

3.A

linear

combination

(function)

of

two

(or

more)

normally

distributionrandom

variables

is

itself

normally

distributed.f

x

x2221e

2【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程+講義37-148Continuous

Probability

Distribution90%95%99%

+

1.65

+

2.58

+

1.96The

confidence

intervalsApproximaApproximaApproximaApproximay

68%

of

all

observations

fall

in

the

interval

μ

±σy

90%

of

all

observations

fall

in

the

interval

μ

±

1.65σy

95%

of

all

observations

fall

in

the

interval

μ

±

1.96σy

99%

of

all

observations

fall

in

the

interval

μ

±

2.58σX

~

N(μ

,

σ2)

-2.58

-

1.65

-1.96【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程+講義38-148Continuous

Probability

DistributionStandard

Normal

DistributionThe

standard

normal

distribution

is

the

normal

distribution

withmean

=

0

and

variance

2

=

1.If

X~N(μ,

σ2),

thenCritical

Z

ValuesExampleX

~

N(70,

9),

compute

the

probability

of

X

≥

75.9Z

=

(75.9

–

70)/3

=

1.96,

P(X

≥

75.9)

=

1

–

97.5%

=

2.5%Compute

64.12

≤

X

≤

75.9；

64.12

≥

X

and

X

≥

75.9?Z

X

~

N0,1Critical

Z

ValueTwo-Side

ConfidenceOne-Sided

Confidence1.64590%95%1.9695%97.5%2.3398%99%2.5899%99.5%【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程

+講義39-148Continuous

Probability

DistributionHow

to

usethe

Z-table【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程+講義40-148Continuous

Probability

DistributionStudent’s

t

distributionIt

is

similar

to

the

normal,

except

it

exhibits

slightly

heavier

tails.It

is

symmetricalIt

has

mean

of

zero.As the

d.f.

increase,

the

t-distribution

converges

with

the

standardnormal

distribution.t

X

Xsx

nBoth

the

normal

and

student’s

t

distribution

characterize

the

samplingdistribution

of

the

sample

mean.

The

difference

is

that

the

normal

is

used

when

we

know

the

population

variance.

The

student’s

t

is

used

when

we

must

rely

on

the

sample

variance.

In

practice,

we

don’t

knowthe

population

variance,

so

the

student’s

t

is

typically

appropriate.【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程+講義41-148Continuous

Probability

DistributionLognormal

DistributionThe

Black-Scholes

Model

assumes

that

the

price

of

the

underlyingasset

is

lognormally

distributed.If

lnX

is

normal,

then

X

is

lognormal;

if

a

variable

is

lognormal,

its

natural

log

is

normal.It

is

useful

for

modeling

asset

priceswhich

never

take

negative

values.Right

skewed.Bounded

from

below

by

zero.【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程+講義42-148+講義【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程Continuous

Probability

DistributionThe

Chi-Square

(χ2)

Probability

DistributionThe

chi-square

test

statistic,

χ2,

with

n-1

degrees

of

freedom,

iscomputed

as:Notice2i(k

)~

2Z

Z2

Z2

Z21

2

kn120n

1s22df=3043-148Properties

of

the

Chi-Square

DistributionThe

chi-square

distribution

take

only

positive

value

and

ranges

from

0

toinfinity

(after

all,

it

is

the

distribution

of

a

squared

ty).The

chi-square

distribution

is

a

positive

skewed

distribution,

the

degree

of

theskewness

depending

on

the

d.f.For

comparatively

few

d.f.

the

distribution

is

highly

skewed

to

the

right,

butas

the

d.f.

increase,

the

distributionand

approaches

the

normal

distribution.E(X)=

k,

D(X)=2k,

where

k

is

the

d.f.es

increasingly

symmetricalIf

Z1

and

Z2

are

two

independent

chi-square

variables

with

k1

and

k2

d.f.,

thentheir

sum

(Z1

+

Z2

)

is

also

a

chi-square

variable

with

d.f.=(k1

+

k2

).【夢軒考資

】

6454842Chi-Square

Distribution專業(yè)提供CFA

FRM全程+講義44-148Continuous

Probability

DistributionF-DistributionIf

U1

and

U2

are

two

independentchi-squareddistributionswith

k1

andk2

degrees

of

freedom,

respectively,

then

X:follows

an

F-distribution

with

parameters

K1

and

K2.As

d.f.

increase,

approaches

normal.The

F

distribution

is

also

called

the

variance

ratio

distribution.

The

Fratio

is

the

ratio

of

sample

variances,

with

the

greater

sample

variancein

the

numerator:1

2~

Fk

,k

2

2k1U

kX

U1Ys2s2F

X【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程

+講義45-148ExerciseOn

a

multiple

choice

exam

with

four

choices

for

each

of

six

questions,what

is

the

probability

that

a

student

gets

less

than

two

questions

correctsimply

by

guessing?0.46%23.73%35.60%53.39%【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程

+講義46-148ExerciseAssume

random

variance

is

normally

distributed

with

mean

of

10

and

avariance

of

25.

Without

using

a

calculator,

what

is

the

probability

that

Xfalls

within

1.75

and

21.65?68%94%95%99%【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程

+講義47-148ExerciseAssume

a

99%

daily

VaR

model

is

perfectly

accurate.

Specifically,

among

100days,

we

expect

a

loss

that

exceeds

VaR

on

exactly

one

day.

Using

a

binomialdistribution,

over

a

series

of

20

trading

days,

which

is

nearest

to

the

probabilitythat

the

daily

loss

will

exceed

VaR

on

exactly

two

days?0.44%0.93%1.59%2.36%The

frequency

of

an

operational

risk

event

type

–

damage

to

physical

assets

–

ischaracterized

by

a

Poisson

distribution.

Over

a age

year,

a

companyexpects

36

of

these

particular

loss

events.

During

the

next

month,

which

isnearest

to

the

probability

the

company

will

experience

exactly

zero

of

theseevents?3.4%5.0%C.

7.5%D.

9.1%【夢軒考資

】6454842

專業(yè)提供CFA

FRM全程+講義48-148ExerciseAssume

the

population

of

hedge

fund

returns

has

an

unknown

distribution

withmean

of

8%

and

volatility

of

10%.

From

a

sample

of

40

funds,

what

is

theprobability

the

sample

mean

return

will

exceed

10.6%?5%8%10%12%Which

of

the

followingexhibit

positively

skewed

distributions?Normal

DistributionLognormal

DistributionThe

Returns

of

Being

Short

a

Put

OptionThe

Returns

of

Being

Long

a

Call

OptionII.onlyIII.onlyII

and

IV

onlyI,

III,

and

IVonly【夢軒考資

】6454842

專業(yè)提供CFA

FRM全程+講義49-148ExerciseWhich

of

the

following

statements

best

characterizes

the

relationshipbetween

the

normal

and

lognormal

distributions?The

lognormal

distribution

is

the

logarithm

of

the

normal

distribution.If

the

natural

log

of

the

random

variable

X

is

lognormally

distributed,then

X

is

normally

distributed.If

X

is

lognormally

distributed,

then

the

natural

log

of

X

is

normally

distributed.The

two

distributions

have

nothing to

do

with

one

another.【夢軒考資

】

6454842

專業(yè)提供CFA

FRM全程

+講義50-148【夢軒考資

】概率與分布常用離散概率分布：二項式分布與泊松分布基本特征與計算？常見連續(xù)概率分布：正態(tài)分布與標(biāo)準(zhǔn)正態(tài)分布基本特征與計算?其他常見分布的基本特征(Student’s

t

distribution;Lognormal

distribution;Chi-Squared

Distribution;

F-distribution)專業(yè)提供CFA

FRM全程+講義51-148【夢軒考資

】tative

ysisProbabilityBasic

StatisticsDistributions4.

Hypothesis

Tests

and

Confidence

IntervalsLinear

RegressionSimulation

ModelingEstimating

Volatilities

and

Correlations專業(yè)提供CFA

FRM全程+講義52-148FrameSampling

and

EstimationPoint

Estimation、Confidence

Interval

EstimateBest

Linear

Unbiased

Estimator

(BLUE)Hypothesis

TestsThe

basis

of

HypothesisThe

application

of

HypothesisTest

of

Single

Population

MeanTest

of

Single

Population

VarianceTest

of

Variances

Difference【夢軒考資

】6454842

專業(yè)提供CFA

FRM全程+講義53-148Statistical

Inference:

Estimation

and

Hypo