BUSS1020 Chapter Notes - Chapter 5: Continuous Or Discrete Variable, Collectively Exhaustive Events, Standard Deviation
28 May 2018
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CHAPTER 5: DISCRETE PROBABILITY DISTRIBUTIONS
THE PROBABILITY DISTRIBUTION FOR A DISCRETE VARIABLE:
• Random variable: represents the possible outcomes from an uncertain event
o Discrete RV: set of all possible outcomes is a finite – integers
o Continuous RV: takes on values at every point over a given interval (can be decimal)
• Expected value of a discrete RV (mean):
o
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§ xi = ith value of discrete variable X
§ P(X = xi) = probability of occurrence of ith value of X
§ E(X) = sum of (X –
%
)2 x P(X)
• Variance and standard deviation of a discrete RV:
COVARIANCE OF A PROBABILITY DISTRIBUTION AND ITS APPLICATION
IN FINANCE:
• Measures the strength of linear relationship b/w two numerical
random variables X and Y
o Higher covariance value = stronger relationship
o Covariance:
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• The Sum of 2 Random Variables:
o E(X):
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o Variance:
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o SD:
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• Linear Transformations
o E(X):
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o Var:
9:;
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:%8@6
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=" :=./
=8@=.0
=8>:@./0
• Portfolio returns:
o E(X) of weighted sum of 2 asset returns:
#2C% 8
$
D3C
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o Variance of weighted sum of 2 asset returns:
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=8>C$D3C&./0
§ a = w and b = 1 – w
BINOMIAL DISTRIBUTION:
• RV X counts number of ‘events of interest’ occurring from a fixed no. of observations (n) à WITH REPLACEMENT
o Properties:
§ Sample has a fixed number of observations, n
§ Two categories are mutually exclusive and collectively exhaustive
§ Probability of each observation is constant à Event of interest = p
§ Observations are independent
• Infinite pop, without replacement
• Finite pop, with replacement
o
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§ Mean:
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§ Variance and standard deviation:
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POISSON DISTRIBUTION:
• Applies when counting the number of times an event occurs in a given “area of opportunity”
o Properties:
§ Probability event occurs in one area of opportunity is the same for all areas of opportunity
§ Events occur independently of each other
§ P(E) in AOO approaches 0 as AOO becomes smaller
o
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E
Q
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§ Mean:
! " Q
§ Variance and standard deviation:
.="Q P . "
W
Q
- l = expected number of events
- e = natural log (2.71828)
- x = number of events observed in AOO
- n = number of observations
- p = probability of event of interest
- x = number of events of interest in the sample
E(X) = 1.4
à
SUM(C2:C8)
SD = SQRT of Var
à
square root
1.428
Var = SUM(E2:E8) = 1.428
Xi = the ith outcome of X
Yi = the ith outcome of Y
(XiYi) = prob of occurrence of ith outcome of X and of Y
