Engineering Statistics - IE 261

Chapter 4Continuous Random Variables andProbability Distributions

URL: http://home.npru.ac.th/piya/ClassesTU.html

http://home.npru.ac.th/piya/webscilab

4-1 Continuous Random Variables

current in a copper wirelength of a machined part

Continuous random variable X

4-2 Probability Distributions and Probability Density Functions

Figure 4-1 Density function of a loading on a long, thin beam.

• For any point x along the beam, the density can be described by a function (in grams/cm)• The total loading between points a and b is determined as the integral of the density function

from a to b.

4-2 Probability Distributions and Probability Density Functions

Figure 4-2 Probability determined from the area under f(x).

4-2 Probability Distributions and Probability Density Functions

Definition

4-2 Probability Distributions and Probability Density Functions

Figure 4-3 Histogram approximates a probability density function.

0P X x because every point has zero width

4-2 Probability Distributions and Probability Density Functions

Because each point has zero probability, one need not distinguish between inequalities such as < or for continuous random variables

Example 4-2

SCILAB:-->x0 = 12.6;-->x1 = 100;-->x = integrate('20*exp(-20*(x-12.5))','x',x0,x1) x = 0.1353353

4-2 Probability Distributions and Probability Density Functions

Figure 4-5 Probability density function for Example 4-2.

Example 4-2 (continued)

SCILAB:-->x0 = 12.5;-->x1 = 12.6;-->x = integrate('20*exp(-20*(x-12.5))','x',x0,x1) x = 0.8646647

4-3 Cumulative Distribution Functions

Definition

4-3 Cumulative Distribution Functions

Example 4-4

4-3 Cumulative Distribution Functions

Figure 4-7 Cumulative distribution function for Example 4-4.

4-4 Mean and Variance of a Continuous Random Variable

Definition

4-4 Mean and Variance of a Continuous Random Variable

Example 4-8