Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Material Type: Notes; Professor: Gerig; Class: Introduction to Mathematical Statistics I; Subject: Statistics; University: North Carolina State University; Term: Unknown 2008;
Typology: Study notes
1 / 6
This page cannot be seen from the preview
Don't miss anything!
Intro duction (4.1)
Probability distribution (4.2)
Exp ected value (4.3)
Uniform distribution (4.4)
Normal distribution (4.5)
Gamma distribution (4.6)
Beta distribution (4.7)
Moments (4.9)
Tchebyshe 's theorem (4.10)
A continuous r.v. takes an uncountably in nite numb ers of values.
The probability distribution for a discrete r.v. can always b e given by assigning a p ositive probability to each of the p ossible values of the variable, and the sum of probability is 1.
Unfortunately, the probability distribution for a continuous r.v. cannot b e sp eci ed in the same way. It is mathematically imp ossible to assign nonzero probabilities to all the p oints on a line interval and the same time satisfy the requirement that the probabilities of the distinct p ossible values sum to 1.
De nition
Let Y denote a r.v. with distribution function F (Y ). Y is said to b e continuous if the distribution function F (y ) is continuous, for 1 < y < 1 :
De nition
Let F (Y ) b e the distribution function for a continous r.v. Y. Then f (y ), given by
f (y ) = dF dy^ (y ) = F 0 (y )
wherever the derivative exists, is called the probability density function for the r.v. Y.
Theorem
Prop erties of a Density Function. If f (y ) is a density function, then
1 f^ (y^ )dy^ =^1
