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

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.

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

 

Example 1.11  

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).

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?