Continuous Probability Distributions

Name

Probability Density Function (pdf), f(x)

Mean, Variance, Moment Generating Function

Comments

Uniform

over (a, b)

 

•

• non-informative, randomness distribution

Gamma

parameters

 

• Very rich family with different shapes

Exponential

parameters

 

• Gamma with n=1

•

(survival function)

• memoryless property

Chi-square

parameter

 

 (nu) is called the degrees of freedom

• Gamma with

Beta

parameters

 

• A good model for proportions (Bayesian inference)

Normal

parameters

 

Standard Normal m=0, s=1

 

• Bell shaped curve

• To find a normal probability use the Table 2.3 on page 81

• If  has  then  has

• Normal approximation to binomial. Let X has Binom(n,q).

Make the continuity correction and use the fact that

Bivariate Normal

Circular normal distribution r=0, s1=s2

• If X and Y have a bivariate normal distribution then

1. Y given X=x has a normal distribution with

2. X given Y=y has a normal distribution with

3. X and Y are independent iff r=0.