ME290M Expert Systems in Mechanical Engineering Spring 1999, T-Th 12:30-2:00 pm 1165 Etcheverry Hall, Course Control No. 56369 http://best.me.berkeley.edu/~aagogino/me290m/s99

Many variables are better modeled as continuous rather than discrete random variables. The representation of continuous random variables as points on a probability wheel is from R.A. Howard , "Probability", Mathematics Associated with System Engineering, MIT Press, pp. 3-48.

1 Wheel of Fortune

The "Wheel of Fortune" is a disk with a freely spinning arrow pivoted at the center.

1.1 Discrete Wheel of Fortune

Assume that the Wheel of Fortune is divided into eight equal sectors. If it is a "fair" wheel, the probability that a spinning arrow will stop in any one of the eight sectors is equally probable with probability of 1/8.

The probability mass function associated with this Wheel of Fortune with equal sectors is shown below.

1.2 Continuous Wheel of Fortune

What is the probability Pr(x) that the spinning arrow on a "fair" Wheel of Fortune stops at any specific value x on the real number line between zero and one? There are an infinite number of possibilities and thus the probability of stopping at any one must be zero. We know from the second axiom of probability (section 10.4) that the probability of all events in the universe must equal one. Thus

In fact the probability that an event is between any interval a · x · b is

There is only one probability density function for the "fair" Wheel of Fortune that will satisfy equation [2] and is numerically the same for all values of x, that is Pr(x=xo ) =1 for all 1 · xo · 1. The probability density function for the fair Wheel of Fortune is shown graphically below.

The following sections on continuous random variables parallels that of discrete variables in Chapter 12. Integral calculus is used instead of discrete summation.

2 Expectation of a Continuous Random Variable

3 Probability Density Function Pr(x)

The probability that the random variable x takes on a continuous value xo is defined by means of the probability density function Pr(x|H) with the notation below:

                          Pr(x) = Pr(x=xo) for -¥< xo <¥

4 Cumulative Probability Distribution Pr(x<y|y)

          Pr(x · y|y) = Pr(x · y|y=yo) for all -¥ < yo < ¥

The following are useful relationships concerning probability density functions:

                                                Pr(x · y|y=-¥) = 0

                                               Pr(x · y|y=¥) = 1

The probability that an event is between any interval a < x < b is

5 Complementary Cumulative Distribution Pr(x>y|y)

6 Unit Impulse

Consider the following probability density function:

7 Moments

Moments of probability density functions are a generalization of the concept of expected value. Moments can be used to define the characteristics of a probability mass function (discrete) or density function (continuous). The quantity mi is the ith moment about the origin.

The Kth moment about the origin of the random variable x is defined as the expected value of xk:

7.1 First Moment: Mean

For k=1,

7.2 Second Moment: Variance about the origin

For k=2,

7.3 Kth Central Moment

Central moments of a random variable x are defined with respect to the mean E(x). The Kth moment about the mean of the random variable x is designated as ck and defined as the expected value of (x - E(x))k :

7.4 First Moment: Mean about the Mean = 0

For k= 1,

7.5 Second Moment: Variance (about the mean is implied)

For k=2,

The standard deviation is the square root of the variance (about the mean is implied) and is designated as s.

7.6 Third Moment about the Mean: Skewness

The third moment is related to the symmetry of the probability density function and is sometimes referred to as the skewness of the probability density function.

The coefficient of skewness "c3" is a dimensionless parameter that is sometimes used to provide a relative measure of the third moment (skewness) with respect to the corresponding power of the second moment (variance):