Transcript MATH408: PROBABILITY & STATISTICS

MATH408: Probability & Statistics
Summer 1999
WEEK 3
Dr. Srinivas R. Chakravarthy
Professor of Mathematics and Statistics
Kettering University
(GMI Engineering & Management Institute)
Flint, MI 48504-4898
Phone: 810.762.7906
Email: [email protected]
Homepage: www.kettering.edu/~schakrav
STUDY OF RANDOM
VARIABLES
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Probability functions
Probability density function (continuous)
Probability mass function (discrete)
Cumulative probability distribution function
Probability Density Function
Example 3.1
Uniform
X = current measured in a thin copper wire (in mA)
The PDF of X is given by f(x) = 0.05, 0  x  20.
Example 3.1 (cont’d)
• Find
– P( X < 8)
– P( X < 8 / X > 6 )
Example 3.2
Exponential
X = diameter (mm) of hole drilled in a sheet metal
component
Example 3.2 (cont’d)
• Find
– P( X > 12.6)
– P( X < 14 / X >12.6 )
3.4.2 Mean and Variance of a Continuous
Random Variable (page 61)
EXAMPLES
NORMAL (GAUSSIAN)
• The most important continuous distribution
in probability and statistics
• The story of the outcome of normal is really
the story of the development of statistics as
a science.
• Gauss discovered this while incorporating
the method of least squares for reducing the
errors in fitting curves for astronomical
observations.
PDF OF NORMAL
Graphs of various normal PDF
ILLUSTRATION OF CALCULATION OF NORMAL
PROBABILITIES- EXAMPLE 3.7
EXAMPLE (cont’d)
(4) P(-1.25 < Z < 0.37)
EXAMPLE (cont’d)
(6) P(Z > ???) = 0.05 or P(Z < ???) = 0.95
How to standardize?
Standardize (cont’d)
EXAMPLES
HOME WORK PROBLEMS
CHAPTER 3
Sections: 3.1 through 3.5
1-9, 10, 15, 18, 21, 22, 23, 27-29, 31
Probability Plots (revisiting)
• Frequently you will be dealing with the
assumption of normal populations.
• Questions: (1) How do we verify this?
• (2) How do we rectify if the assumption is
violated?
• To answer (1), we look at probability plot
(normal probability plot).
• For (2), we use transformation.
Plots(cont’d)
• Construction of a probability plot
can be done in two ways. First
calculate the percentiles of the data
points, say x(j).
1. On the probability paper (which will
have the percentiles along the y-axis
and the values of the data along the xaxis) plot the values, x(j),.
Plots(cont’d)
2. Calculate y(j), the corresponding
percentiles of the probability
distribution. Plot (x(j),y(j),) on a regular
paper.
If the points lie pretty much on a
straight line, then we can conclude that
there is no evidence to refute the
assumption.