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Discreet and continuous probability
- 1. DISCREET AND CONTINUOUS PROBABILITY PRESENTEDBY: Noopur Joshi MSc. I SEM DEPARTMENT: BIOTECHNOLOGY
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- 3. • Probability isa measure of the expectation that an event will occur or a statement is true. • Probabilities are given a value between 0 (will not occur) and 1 (will occur). • The higher the probability of an event, the more certain we are that the event will occur.
- 4. • The concepthas been given an axiomatic mathematical derivation in probability theory, which is used widely in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence/machine learning and philosophy to, for example, draw inferences about the expected frequency of events. • Probability theory is also used to describe the underlying mechanics and regularities of complex systems.
- 5. Etymology • The wordProbability derives from the Latin probabilitas, which can also mean probity, a measure of the authority of a witness in a legal case in Europe, and often correlated with the witness's nobility. • In a sense, this differs much from the modern meaning of probability, which, in contrast, is a measure of the weight of empirical evidence, and is arrived at from inductive reasoning and statistical inference.
- 6. Interpretations • When dealingwith experiments that are random and well-defined in a purely theoretical setting (like tossing a fair coin), probabilities describe the statistical number of outcomes considered divided by the number of all outcomes (tossing a fair coin twice will yield HH with probability 1/4, because the four outcomes HH, HT, TH and TT are possible).
- 7. • When itcomes to practical application, however, the word probability does not have a singular direct definition. • There are two major categories of probability interpretations, whose adherents possess conflicting views about the fundamental nature of probability:
- 8. • Objectivists assignnumbers to describe some objective or physical state of affairs. • The most popular version of objective probability is frequentist probability, which claims that the probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. • This interpretation considers probability to be the relative frequency "in the long run" of outcomes. • A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, even if it is performed only once.
- 9. • Subjectivists assignnumbers per subjective probability, i.e., as a degree of belief. • The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as experimental data to produce probabilities. • The expert knowledge is represented by some (subjective) prior probability distribution. The data is incorporated in a likelihood function. • The product of the prior and the likelihood, normalized, results in a posterior probability distribution that incorporates all the information known to date.
- 10. Probability Distributions: Discrete vs.Continuous • All probability distributions can be classified as discrete probability distributions or as continuous probability distributions, depending on whether they define probabilities associated with discrete variables or continuous variables
- 11. Discrete vs. ContinuousVariables • If a variable can take on any value between two specified values, it is called a continuous variable; otherwise, it is called a discrete variable
- 12. • Suppose thefire department mandates that all fire fighters must weigh between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire fighter's weight could take on any value between 150 and 250 pounds. • Suppose we flip a coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. However, it could not be any number between 0 and plus infinity. We could not, for example, get 2.5 heads. Therefore, the number of heads must be a discrete variable.
- 13. Just like variables, probabilitydistributions can be classified as discrete or continuous.
- 14. Discrete Probability Distributions •DEFINITION: If a random variable is a discrete variable, its probability distribution is called a discrete probability distribution.
- 15. EXAMPLE • Suppose youflip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. • Now, let the random variable X represent the number of Heads that result from this experiment • The random variable X can only take on the values 0, 1, or 2, so it is a discrete random variable.
- 16. • The probabilitydistribution for this statistical experiment appears below • The above table represents a discrete probability distribution because it relates each value of a discrete random variable with its probability of occurrence Number of heads Probability 0 0.25 1 0.50 2 0.25
- 17. Continuous Probability Distributions •DEFINITION: If a random variable is a continuous variable, its probability distribution is called a continuous probability distribution.
- 18. • A continuousprobability distribution differs from a discrete probability distribution in several ways: 1. The probability that a continuous random variable will assume a particular value is zero. 2.As a result, a continuous probability distribution cannot be expressed in tabular form. 3.Instead, an equation or formula is used to describe a continuous probability distribution.
- 19. • Most often,the equation used to describe a continuous probability distribution is called a probability density function. • Sometimes, it is referred to as a density function, a PDF, or a pdf. For a continuous probability distribution, the density function has the following properties:
- 20. • Since thecontinuous random variable is defined over a continuous range of values (called the domain of the variable), the graph of the density function will also be continuous over that range. • The area bounded by the curve of the density function and the x-axis is equal to 1, when computed over the domain of the variable. • The probability that a random variable assumes a value between a and b is equal to the area under the density function bounded by a and b.
- 21. • For example,consider the probability density function shown in the graph below: Suppose we wanted to know the probability that the random variable X was less than or equal to a. The probability that X is less than or equal to a is equal to the area under the curve bounded by a and minus infinity - as indicated by the shaded area.
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- 23. • Note: Theshaded area in the graph represents the probability that the random variable X is less than or equal to a. This is a cumulative probability. However, the probability that X is exactly equal to a would be zero. A continuous random variable can take on an infinite number of values. The probability that it will equal a specific value (such as a) is always zero.