Chapter 2 Continuous Distributions - Bauer College of Business

RS ¨C 2 ¨C Continuous Distributions

Chapter 2

Continuous Distributions

(for private use, not to be posted/shared online)

Continuous random variables: Review

? For a continuous random variable X the probability distribution is

described by the probability density function f(x), which has the

following properties :

1. ? ? ¡Ý 0.

? ? ??

2.

3. ? ?

? ? ??

?

1.

1.

?

? ? ??

??

?

?

? ? ??

1

RS ¨C 2 ¨C Continuous Distributions

Continuous random variables: Review

? Continuous distributions allow for a more elegant mathematical

treatment.

? They are especially useful to approximate discrete distributions.

Continuous distributions are used in this way in most economics and

finance applications, both in the construction of models and in

applying statistical techniques.

? Most used continuous distributions:

- Uniform

- Normal

- Exponential

- Weibull

- Gamma

The Uniform distribution from a to b

Abraham de Moivre (1667-1754)

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RS ¨C 2 ¨C Continuous Distributions

Uniform Distribution

? A random variable, X, is said to have a Uniform distribution from a to b if

X is a continuous RV with probability density function f(x):

f

?x

?

1

?

?

? ? b ? a

??

0

a ? x ? b

o th e rw is e

? PDF: Uniform distribution (from a to b)

0 .4

0 .3

0 .2

0 .1

0

0

5

10

15

Uniform Distribution: CDF

? The CDF, F(x), of the uniform distribution from a to b:

x?a

? 0

?x?a

?

F ? x ? ? P ? X ? x? ? ?

?b ? a

?? 1

0.4

f ? x?

0.3

a? x?b

x?b

0.2

F ? x?

0.1

0

0

a

5

x

b

10

15

3

RS ¨C 2 ¨C Continuous Distributions

Uniform Distribution function: Graph of F(x)

? 0

?x?a

?

F ? x ? ? P ? X ? x? ? ?

?b ? a

?? 1

F ? x?

x?a

a? x?b

x?b

1

0.5

0

0

5

10

15

a

b

x

Uniform Distribution: Comments

? When a=0 and b=1, the distribution is called standard uniform

distribution, which is a special case of the Beta distribution, with

parameters (1,1)

? The uniform distribution is not commonly found in nature (but very

common in casinos!). Because of its simplicity, it is used in theoretical

work (for example, signals/private information are draw from a UD). It

is used as prior distribution for binomial proportions in Bayesian

statistics.

? It is also particularly useful for sampling from arbitrary distributions. A

general method is the inverse transform sampling method, which uses the

CDF of the target RV. This method is very useful in theoretical work.

Since simulations using this method require inverting the CDF of the

target variable, alternative methods have been devised for the cases

where the CDF is not known in closed form.

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RS ¨C 2 ¨C Continuous Distributions

The Normal distribution

(Also called Gaussian or Laplacian)

Abraham de Moivre (1667-1754)

Pierre-Simon, m. de Laplace (1749¨C1827)

Carl Friedrich Gauss (1777¨C1855)

The Normal distribution: PDF

A random variable, X, is said to have a normal distribution with mean ?

and standard deviation ? if X is a continuous random variable with

probability density function f(x):

? ?

1

2??

exp

?

?

2?

5

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