Common Probability Distributions

Discrete Uniform Distribution

Binomial Distribution
o
Bernoulli Random Variable:
A Bernoulli trial produces one of two
outcomes. A binomial random variables is defined as the number of
successes in n Bernoulli trials.
o
The binomial distribution makes the following assumptions: the
probability of success is constant for all trials and the trials are
independent.
o
A Binomial random variable is completely described by two parameters n
and p. A Bernoulli random variable is a binomial random variable with
n=1.
o
Probability of x successes in n trials is given by
(1
)
x
n
x
n
p
p
x


÷
o
Do Sample problems from the curriculum
o
Bernoulli, B(1,p) : Mean = p; Variance = p(1p)
o
Binomial, B(n,p); Mean=np; Variance = np(1p)

Continuous Random Variables
o
The pdf for a uniform random variable is
1
( )
;0
f x
fora
x
b
otherwise
b
a
=
<
<

o
For a continuous uniform random variable: Mean= (a+b)/2; Variance= (b
a)
2
/12

Normal Distribution
o
Central Limit Theorem: Sum (and mean) of a large number of independent
random variables is approximately normally distributed.
o
Safety First Optimal Portfolio: If returns are normally distributed the
safetyfirst optimal portfolio maximizes the safety first ratio (similar to
Sharpe Ratio). For a portfolio with a given SFRatio, the probability that its
return will be less than the threshold is N(SFRatio).

Lognormal Distribution
o
A random variable Y follows a lognormal distribution if its natural log is
normally distributed. The two parameters of lognormal distribution are the
mean and standard deviation of its associated normal distribution.
o
Mean = exp(μ+.5σ
2
); Variance=exp(2μ+σ
2
)x[exp(σ
2
)1]
o
If a stocks continuously compounded return is normally distributed, then
future stock price is necessarily lognormally distributed.
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 Fall '10
 Roberts
 Normal Distribution, Probability theory, binomial random variable, Bernoulli random variable, Lognormal Distribution

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