Probability & Statistics for Engineers & Scientists, byWalpole, Myers, Myers & Ye ~Chapter 6 Notes Class notes for ISE 201 San Jose State University Industrial & Systems Engineering Dept. Steve Kennedy
Continuous Uniform Distribution • The continuous uniform distribution is the simplest of the continuous distributions. It models the situation where the probability is uniform in the interval [A,B]. • The density function is • This density function is a rectangle with base B - A. The height of the rectangle is? • The height is 1 / (B - A).
Uniform Distribution • The mean and variance of the uniform distribution are and
Normal Distribution Examples • The normal distribution is the most important and widely used distribution in statistics. Some examples: • This bell-shaped curve, sometimes called the Gaussian distribution, explains many natural phenomena. • Average age of the world's population. • Physical measurements like blood pressure. • Standardized test scores. • Average precipitation levels. • Average price of certain stocks in the stock market. • The thickness of metal plates from a factory (can't be negative but still a good approximation since tail isn't close to 0). • The Central Limit Theorem in statistics indicates that random variables that are the sum of a number of component variables follow the normal distribution.
Normal Distribution • The density function for a normally distributed random variable with mean and standard deviation , is • As before, determines the center of the bell-shaped curve, while determines the spread. Notes: • The curve is symmetric about the mean. • As expected, the the total area under the curve equals 1. However, the function cannot be integrated in closed form. • The mode (where the curve is at a maximum) is at x = . • The curve approaches the y axis as x moves away from . • The curve has a point of inflection at x = .
Standard Normal Distribution • Since it's not feasible to publish tables for every possible value of and , we define the standard normal distribution to be a normal distribution with = 0 and = 1, and publish a table for that. • Then, every normal probability question can be reduced to an equivalent question about the standard normal distribution. • A standard normal random variable can be from any normal random variable with a simple transformation:Intuitively, Z shows the number of standard deviations above or below (if negative) the mean.
Using the Normal Distribution Table (A.3) • The table shows F(z), which is? • The cumulative standard normal distribution. Which means? • We use the key fact that the normal distribution is symmetric around to mean, and that the area under the curve equals 1, to find any desired probability. • To use the table, first convert the normal random variable to a standard normal random variable. • Ex: If a normal random variable X has = 50 and = 25, find P(X > 12). • Converting to standard normal, we have P(X > 12) = P(Z > -1.52). (12 is 1.52 standard deviations below the mean.) • By table A.3, F(-1.52) = .0643. So P(Z > -1.52) = 1 - .0643 = .9357. • Or, by symmetry, P(Z > -1.52) = P(Z < +1.52) = .9357. • So, P(X > 12) = .9357.
More Normal Distribution Calculations • Use properties of a cumulative distribution to calculate the probability of being within a given range. • Ex: If a normal random variable X has = 50 and = 25, find P(62 > X > 32). First step? • Convert to standard normal. • P(62 > X > 32) = P(0.48 > Z > -0.72). Next? • P(0.48 > Z > -0.72) = P(Z 0.48) - P(Z -0.72) = ? • P(Z 0.48) - P(Z -0.72) = 0.6844 - 0.2358 = .4486. • Reverse calculation: starting with a probability, find X. • Using and above, what values of X centered above and below the mean have p = .80 of falling between them. First? • Find Z values with 80% of the area between them. z = 1.28. • Next convert from standard normal back to X. • So, the limits are x = 50 1.28(25) = (18, 82).
Distribution Approximation Notes • Continuity correction: Suppose a population of resistors follows a normal distribution, but that resistance measurements are in integer values of ohms. • Here we assume that any resistance from 42.5 to 43.5 = 43. • So the probability of any outcome can be calculated using the normal distribution. • This example uses a continuity correction to approximate a discrete distribution with a continuous distribution. • If n is large and p is very small (or very large), the binomial distribution can be approximated by what? • The Poisson distribution. • What is and for the binomial distribution? • = np. • 2 = npq.
Normal Approximation to Binomial • If X binomial with parameters n and p, then for n . This is a good approximation: • If n is large and p is not too close to 0 or 1. • Or, if p is reasonably close to 1/2, even when n is small. • To find the probability that x = k? • Use the normal distribution for the area x = k (1/2). • For k = 0, we use P(x +0.5) so probability will sum to 1. • Approximation works well if both np and nq 5. • The larger n is, the better the approximation will be.
Gamma Distribution • The gamma distribution with parameters and , is defined aswhere > 0, > 0, and • Note that if is an integer, () simplifies to (-1) ! thenand
Exponential Distribution • An exponential distribution with parameter , is defined asthenand • Exponential distribution notes: • The mean equals the standard deviation. • Note that = (1/), where is the parameter used in the book for the exponential distribution.
Exponential and Gamma Distributions • The cumulative distribution for the exponential is very simple. It is • It is easiest to use the cumulative distribution to calculate exponential distribution probabilities. • The gamma distribution for an integer (also called the Erlang distribution) is shown below.
Exponential and Poisson Distributions • For a Poisson process, if is the mean event arrival rate, what is the distribution of the number of events occurring in a time interval of length t? • It is a Poisson distribution with mean t. • It can also be shown for a Poisson process that the distribution of the time until the next event occurs is exponential with parameter . • Similarly, an Erlang distribution with parameter is the distribution of the time until events occur. • Exponential distribution is a gamma distribution with = 1. • There are many important applications of the exponential distribution (and the gamma distribution) in the areas of reliability and queuing theory.
Exponential Dist'n - Memoryless Property • What was the memoryless property for a Poisson process? • The number of occurrences in one time interval is independent of the number in any other disjoint time interval. • Thus future arrivals are completely independent of the past history of arrivals. • How does the memoryless concept apply to the exponential distribution of the time between arrivals? • The memoryless property of the exponential distribution implies that • Or, the probability of waiting longer than t time units for the next arrival is independent of how long we've already waited.
Lognormal Distribution • Lognormal distribution. For some applications, the natural log of the observations follows a normal distribution. Here and are specified in terms of the logs of the original data. • Then z =(ln(x) - )/ follows a standard normal distribution. • Working backwards, how do we convert back to x? • To convert back to x, we use x = e + z. • The mean and the variance of the lognormal distribution (in the units of the original data) are:
Other Continuous Distributions • Chi-Squared distribution. Used for statistical inference. We will use the chi-squared table in the appendix later in the semester. • Weibull distribution. Has many applications in reliability and life testing and is used to model failure rate. We're skipping this distribution this semester. • The failure rate at time t for the Weibull distribution is • For = 1 failure rate is constant (memoryless). • For > 1 failure rate increases over time. • For < 1 failure rate decreases over time.