Discrete and continuous probability models
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Discrete and continuous probability models Presentation Transcript

  • 1. Discrete and Continuous Probability Models Akshay Kr Mishra-100106039 Sharda University, 4th yr ;ME
  • 2. Probability Distribution? • A probability distribution is a mathematical model that relates the value of the variable with the probability of occurrence of that value in the population. • There are 2 types of probability Distribution- 1. Continuous Probability Distribution 2. Discrete Probability Distribution.
  • 3. Continuous distributions- When the variable being measured is expressed on a continuous scale, its probability distribution is called a continuous distribution. Ex- The probability distribution of metal layer thickness is continuous. Discrete distributions. When the parameter being measured can only take on certain values, such as the integers 0, 1 etc. the probability distribution is called a discrete distribution. Ex- distribution of the number of nonconformities or defects in printed circuit boards would be a discrete distribution
  • 4. Some Imp. Terms • Mean- The Mean of a probability distribution is a measure of the central tendency in the distribution, or its location. • Variance- The scatter, spread, or variability in a distribution is expressed by the variance. • Standard Deviation- The standard deviation is a measure of spread or scatter in the population expressed in the original terms.
  • 5. Types Of Discrete Distribution • Hyper geometric Distribution- An appropriate probability model for selecting a random sample of n items without replacement from a lot of N items of which D are nonconforming or defective. • In these applications, x usually is the class of interest and then that x is the hyper geometric random variable.
  • 6. • Binomial Distribution- Lets consider a process of ‘n’ independent trials. • When the outcome of each trial is either a “success” or a “failure,” the trials are called Bernoulli trials. • If the probability of “success” on any trial—say, p—is constant, then the number of “successes” x in n Bernoulli trials has the binomial distribution.
  • 7. • The binomial distribution is used frequently in quality engineering. • It is the appropriate probability model for sampling from an infinitely large population, where p represents the fraction of defective or nonconforming items in the population. • In these applications, x usually represents the number of nonconforming items found in a random sample of n items.
  • 8. Poisson’s Distribution • We note a Important fact here and that is the mean and variance of the Poisson distribution are both equal to the parameter Lambda.
  • 9. • A typical application of the Poisson distribution in quality control is as a model of the number of defects or nonconformities that occur in a unit of product. • In fact, any random phenomenon that occurs on a per unit (or per unit area, per unit volume, per unit time, etc.) basis is often well approximated by the Poisson distribution. • It is possible to derive the Poisson distribution as a limiting form of the binomial distribution. • That is, in a binomial distribution with parameters n and p, if we let n approach infinity and p approach zero in such a way that np = lambda is a constant, then the Poisson distribution results.
  • 10. Types of Continuous Distribution • Lognormal Distribution-
  • 11. • The lifetime of a product that degrades over time is often modelled by a lognormal random variable. For example-the lifetime of a semiconductor laser. • However, because the lognormal distribution is derived from a simple exponential function of a normal random variable, it is easy to understand and easy to evaluate probabilities.
  • 12. • Normal Distribution- The normal distribution is probably the most important distribution in both the theory and application of statistics. • If x is a normal random variable, then the probability distribution of x is defined as follows.
  • 13. • The normal distribution is used so much that we frequently employ a special notation, to imply that x is normally distributed with mean and variance. • The visual appearance of the normal distribution is a symmetric, unimodal or bell-shaped curve. Area Under Normal Distribution.
  • 14. • The Normal Distribution has many useful properties and one which has found its world wide use is the “Central Limit Theorem”. • Central Limit Theorem- The central limit theorem implies that the sum of n independently distributed random variables is approximately normal, regardless of the distributions of the individual variables. • The approximation improves as n increases.
  • 15. • Exponential Distribution-
  • 16. Area Under the Exponential Distribution The exponential distribution is widely used in the field of reliability engineering as a model of the time to failure of a component or system. In these applications, the parameter is called the failure rate of the system, and the mean of the distribution is called the mean time to failure.
  • 17. THANK YOU!!! :D