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Probability & Statistics for Engineers & Scientists, by Walpole, Myers, Myers & Ye ~ Chapter 6 Notes. Class notes for ISE 201 San Jose State University Industrial & Systems Engineering Dept. Steve Kennedy. Continuous Uniform Distribution.

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probability statistics for engineers scientists by walpole myers myers ye chapter 6 notes
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
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
Uniform Distribution
  • The mean and variance of the uniform distribution are

and

normal distribution examples
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
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
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
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
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
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
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
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
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
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
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
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
  • 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
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.