Discrete and continuous probability models
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This document discusses different types of probability distributions used in statistics. There are two main types: continuous and discrete distributions. Continuous distributions are used when variables are measured on a continuous scale, while discrete distributions are used when variables can only take certain values. Some important continuous distributions mentioned are the normal, lognormal, and exponential distributions. Important discrete distributions include the binomial, hypergeometric, and Poisson distributions. Key terms like mean, variance, and standard deviation are also defined. Examples are provided to illustrate how these probability distributions are applied in fields like quality control and reliability engineering.
Discrete and continuous probability models
- 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.
- 7. • 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.
- 9. • 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.
- 10. 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.
- 11. • 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.
- 12. Types of Continuous Distribution • Lognormal Distribution-
- 13. • 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.
- 14. • 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.
- 15. • 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.
- 16. • 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.
- 18. 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.
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