Notesale: Turn your study into money

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Xi = the ith outcome of X
Y = discrete random variable Y
Yi = the ith outcome of Y
P(X=Xi,Y=Yi) = probability of occurrence of the
ith outcome of X and the ith outcome of Y

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...

Investment
Economic Condition
Prob
...
2

Recession

- $25

- $200

0
...
3

Expanding Economy

+ $100

+ $350

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
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2) +(50)(
...
3) = 50
E(Y) = μY = (-200)(
...
5) + (350)(
...
00 return
and fund Y is averaging a $95
...


Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...
2) + (50 − 50) 2 (
...
3)
= 43
...
2) + (60 − 95) 2 (
...
3)
= 193
...


Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
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2) + (50 − 50)(60 − 95)(
...
3)
= 8,250
Interpretation: Since the covariance is large and
positive, there is a positive relationship between the
two investment funds, meaning that they will likely
rise and fall together
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Chap 5-16

The Sum of
Two Random Variables


Expected Value of the sum of two random variables:

E(X + Y) = E( X) + E( Y )


Variance of the sum of two random variables:

Var(X + Y) = σ 2X+ Y = σ 2X + σ 2Y + 2σ XY


Standard deviation of the sum of two random variables:

σ X + Y = σ 2X + Y
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...




Investment Objective: Maximize return (mean)
while minimizing risk (standard deviation)
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Chap 5-18

Portfolio Expected Return
and Portfolio Risk


Portfolio expected return (weighted average
return):

E(P) = w E( X) + (1 − w ) E( Y )


Portfolio risk (weighted variability)

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

w = proportion of portfolio value in asset X
(1 - w) = proportion of portfolio value in asset Y

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
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30
μY = 95 σY = 193
...
4 (50) + (0
...
4) 2 (43
...
6) 2 (193
...
4)(0
...
30
The portfolio return and portfolio variability are between the values
for investments X and Y considered individually
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
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5

Discrete
Probability
Distributions

Continuous
Probability
Distributions

Binomial

Normal

Poisson

Uniform

Hypergeometric
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...
6

Exponential
Chap 5-21

Binomial Probability Distribution


A fixed number of observations, n




e
...
, 15 tosses of a coin; ten light bulbs taken from a

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