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Formulae Sheet...

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Formula Sheet 1. Descriptive Statistics Sample Mean:

´x =

∑ xi n

∑ xi

Population Mean:

μ=

Sample Variance:

2 ( x i−´x ) ∑ x i2−n x´ 2 ∑ = s=

N

2

Population Variance:

n−1

σ

2

n−1

2 ( x i−μ ) ∑ x 2i −N μ2 ∑ = =

N

N

The standard deviation is the positive square root of the variance. Quartiles: -

n+1 4 n+1 Second Quartile Position: L2 = 2 3 ( n+1 ) Third Quartile Position: L3= 4

First Quartile Position:

L1 =

Quartile Rules: -

Rule 1: If the result is an integer, then the quartile is equal to the ranked value. Rule 2: If the result is a fractional half, then the quartile is equal to the mean of the corresponding values. Rule 3: If the result is neither an integer nor fractional half, round the result to the nearest integer and select the ranked value.

Coefficient of Variation:

σ CV = × 100 % μ

Standardized Value or Z-Score:

Z=

X−μ σ

or

s CV = × 100 % x´

and for a given Z-Value:

X =Zσ + μ

Empirical Rule: The interval of values one standard deviation either above or below the mean, ´x ± 1⋅ s , contains approximately 68% of the items or people within the sample.

2. Probability Theory If N is the total number of opportunities for the event to occur and x is the time of events the event A has occurred:

P ( A )=

x N

Marginal Probability: P ( A )=P ( A ∩B )+ P (A ∩B ' ) where exclusive and collectively exhaustive events. Addition Law:

B and B '

are mutually

P ( A ∪ B ) =P ( A ) + P ( B )−P ( A ∩ B )

¿ P ( A )+ P ( B ) Conditional Probability:

if A and B are mutually exclusive.

P (A ∩B) P ( A ) ⋅ P(B∨A ) = P(B) P(B) ¿ P( A) if A and B are independent

P ( A|B ) =

Statistical Independence: A and B are independent, if and only if

P ( A|B ) =P( A)

P ( A ∩ B) =P ( B) ⋅ P ( A |B )=P ( A ) ⋅P (B ∨ A )

Multiplication Rule:

¿ P ( A )⋅ P(B) if A and B are independent Bayes Theorem:

P ( B|A ) =

P (A ∨B)⋅ P (B ) P (A ∩B) = P( A) P ( A ∨B )⋅P (B )+ P ( A∨B' )⋅ P (B' )

3. Price Indexes Simple Weighted Index:

It =

Pt ×100 % P0

Unweighted Aggregate Price Index:

Laspeyres Price Index:

Paasche Price Index:

It =

∑ Pt × 100 % ∑ P0

∑ Pt ⋅Q0 ×100 % ∑ P0 ⋅Q0 ∑ Pt ⋅Q t × 100 % It = ∑ P0 ⋅Qt It =

4. Probability Distributions Mean or Expected Value of Discrete Distribution: Variance of a Discrete Distribution:

σ2X=∑ ( x i−μ X ) 2 ⋅ P ( X=x i )=∑ x2i ⋅ P ( X=x i ) −μ2X

Standard Deviation of a Discrete Distribution: Covariance:

σ X =√ σ X 2

σ XY =∑ ( x i−E (X ) ) ⋅ ( y i−E ( Y )) ⋅ P(x i y i)=∑ ( x i ⋅ y i ⋅ P ( x i y i) ) −E( X ) ⋅ E (Y )

Where x i y i is the outcome where both variables X and Y respectively. Correlation:

μ X =E( X ) =∑ xi ⋅ P(X =x i)

r=

σ XY

x i and

yi

occur for the discrete random

, ranges between -1 and 1

σ X σY

Portfolio Expected Return and Portfolio Risk:

E ( P )=wE ( X )+(1−w ) E ( Y )

-

Expected Return:

-

Portfolio Risk:

-

w is portion of portfolio value in asset X, (1-w) is portion of portfolio value in asset Y

Binomial Distribution: -

2 2 2 σ P=√ w σ X+ ( 1−w ) σ Y +2 w (1−w ) σ XY 2

P ( X=x )=

n! n −x x p ⋅( 1− p ) x ! ( n−x ) !

Mean or Expected Value: μ=n⋅ p Variance of Binomial: σ 2 =n ⋅ p ⋅(1− p ) ; the standard deviation is the square root of the variance.

Uniform Distribution:

μ= E ( X )=

a+b 2

-

Mean:

-

Standard Deviation:

σ=

where X is a random variable between a and b

b−a √ 12

Normal Distribution: -

Convert the X to a Z-Score Standard Normal Table (below) refers ONLY to If you need to solve Pr ( Z>z ) o Recall Pr ( Z> z ) =1−Pr (Z < z )

-

If you need to solve Pr ( −z < Z< z ) o Recall Pr ( −z tα /2 , n−1

or

or

za

Pr(Z> z )

z> z α

t> tα , n−1

H 0: μ ≥ a H 1 : μ...