1. Lecture 2 Section 1.3 Objectives: Continuous Distributions
Density Function
Discrete Distributions
Mass Function

2. Describing Distributions
Distribution of a variable tells us what values it takes and how often it takes these values.
Consider a population and a sample.
It is often possible to give a concise mathematical description of how the possible values of a variable are distributed or dispersed along the number line or measurement scale.

3. Continuous Distributions
Note that there is no area under the density curve associated with a single value, so :
proportion of x values satisfying a ≤ x ≤ b = proportion of x values satisfying a < x < b

5. Example 1.11

6. Discrete Distributions
Definition
A population or process distribution for a discrete variable x is specified by a mass function p(x) satisfying,
Note that
Proportion of x values between a and b (inclusive)
= p(a) + p(a+1) + … + p(b) for a < b.
Also, proportion of x values satisfying a ≤ x < b
= p(a) + p(a+1) + … + p(b-1).
proportion of x values satisfying a < x ≤ b
= p(a+1) + p(a+2) + … + p(b).
proportion of x values satisfying a < x < b
= p(a+1) + p(a+2) + … + p(b-1).

7. Example
Consider that there is a package of five electric components of a particular type and let x denote the number of satisfactory electric components in the package.
One possible values of x are 0, 1, 2, 3, 4, and 5 . One reasonable distribution for x is
What is the proportion of all packages containing two “good” components?
What is the proportion of packages with at least two “good” components?