Chapter 4 Continuous Random Variables and their Probability Distributions The Theoretical Continuous Distributions starring The Rectangular The Normal The Exponential and The Weibull Chapter 4B

Continuous Uniform Distribution A continuous RV X with probability density function has a continuous uniform distribution or rectangular distribution a b

4-5 Continuous Uniform Random Variable Mean and Variance

Using Continuous PDF’s Given a pdf, f(x), a <= x <= b and and a <= m < n <= b P(m <= x <= n) =

Problem 4-33

Let’s get Normal Most widely used distribution; bell shaped curve Histograms often resemble this shape Often seen in experimental results if a process is reasonably stable & deviations result from a very large number of small effects – central limit theorem. Variables that are defined as sums of other random variables also tend to be normally distributed – again, central limit theorem. If the experimental process is not stable, some systematic trend is likely present (e.g., machine tool has worn excessively) a normal distribution will not result.

4-6 Normal Distribution Definition

4-6 Normal Distribution

The Normal PDF

Normal IQs

4-6 Normal Distribution Some useful results concerning the normal distribution

Normal Distributions

Standard Normal Distribution A normal RV with is called a standard normal RV and is denoted as Z. Appendix A Table III provides probabilities of the form P(Z < z) where You cannot integrate the normal density function in closed form. Fig Standard Normal Probability Function

Examples – standard normal P(Z > 1.26) = 1 – P(Z  1.26) = = P(Z < -0.86) = P(Z > -1.37) = P(z < 1.37) = P(-1.25< Z<0.37) = P(Z<.0.37) – P(Z<-1.25) = = P(Z < -4.6) = not found in table; prob calculator = P(Z > z) = 0.05; P(Z < z) =.95; from tables z  1.65; from prob calc = P(-z < Z < z) = 0.99; P(Z<z) =.995; z = 2.58

Converting Normal RV’s to Standard Normal Variates (so you can use the tables!) Any arbitrary normal RV can be converted to a standard normal RV using the following formula: After this transformation, Z ~ N(0, 1) the number of standard deviations from the mean

4-6 Normal Distribution To Calculate Probability

Converting Normal RV’s to Standard Normal Variates (an example) For example, if X ~ N(10, 4) To determine P(X > 13): from Table III

Converting Normal RV’s A scaling and a shift are involved.

More Normal vs. Std Normal RV X ~ N(10,4)

Example 4-14 Continued (sometimes you need to work backward Determine the value of x such that P(X  x) = 0.98 X ~ N(10,4)

Check out this website An Illustration of Basic Probability: The Normal Distribution See the normal curve generated right in front of your very own eyes

4-8 Exponential Distribution Definition

The Shape of Things

The Mean, Variance, and CDF table of definite integrals

What about the median?

Next Example Let X = a continuous random variable, the time to failure in operating hours of an electronic circuit f(x) = (1/25) e -x/25 F(x) = 1 - e -x/25  = 1/ = E[X] = 25 hours median = (25) = hours  2 = V[X] = 25 2  = 25

Example What is the probability there are no failures for 6 hours? What is the probability that the time until the next failure is between 3 and 6 hours?

Exponential & Lack of Memory Property: If X ~ exponential This implies that knowledge of previous results (past history) does not affect future events. An exponential RV is the continuous analog of a geometric RV & they both share this lack of memory property. Example: The probability that no customer arrives in the next ten minutes at a checkout counter is not affected by the time since the last customer arrival. Essentially, it does not become more likely (as time goes by without a customer) that a customer is going to arrive.

Proof of Memoryless Property A – the event that X t 1 Chapter Two stuff!

Exponential as the Flip Side of the Poisson If time between events is exponentially distributed, then the number of events in any interval has a Poisson distribution. N T events till time T Time between events has exponential distribution Time T Time 0

Exponential and Poisson Let X(t) = the number of events that occur in time t; assume X(t) ~ Pois( t) then E[X(t)] = t Let T = the time until the next event; assume T ~ Exp( ) then E[T] = 1/

4-10 Weibull Distribution Definition

The PDF in Graphical Splendor Beta Delta = 2

More Splendor Delta Beta = 1.5

4-10 Weibull Distribution

The Gamma Function fine print: easier method is to use the prob calculator

4-10 Weibull Distribution Example 4-25

The Mode of a Distribution a measure of central tendency

The Mode of a Distribution

A Weibull Example The design life of the members used in constructing the roof of the Weibull Building, a engineering marvel, has a Weibull distribution with  = 80 years and  = 2.4.

Other Continuous Distributions Worth Knowing Gamma Erlang is a special case of the gamma Used in queuing analysis Beta Like the triangular – used in the absence of data Used to model random proportions Lognormal used to model repair times (maintainability) quantities that are a product of other quantities (central limit theorem) Pearson Type V and Type VI like lognormal – models task times

Picking a Distribution We now have some distributions at our disposal. Selecting one as an appropriate model is a combination of understanding the physical situation and data-fitting Some situations imply a distribution, e.g. arrivals  Poisson process is a good guess. Collected data can be tested statistically for a ‘fit’ to distributions.

Next Week – Chapter 5 Double our pleasure by considering joint distributions.