DISCREET AND CONTINUOUS PROBABILITY

PRESENTED BY:Noopur JoshiMSc. I SEM

DEPARTMENT:BIOTECHNOLOGY

PROBABILITYAN INTRODUCTION

• Probability is a 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.

• The concept has 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.

Etymology

• The word Probability 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.

Interpretations

• When dealing with 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).

• When it comes 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:

• Objectivists assign numbers 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.

• Subjectivists assign numbers 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.

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

Discrete vs. Continuous Variables

• 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

• Suppose the fire 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.

Just like variables, probability distributions can be classified as discrete or

continuous.

Discrete Probability Distributions

• DEFINITION:If a random variable is a discrete variable, its

probability distribution is called a discrete probability distribution.

EXAMPLE

• Suppose you flip 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.

• The probability distribution 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

Continuous Probability Distributions

• DEFINITION:If a random variable is a continuous variable,

its probability distribution is called a continuous probability distribution.

• A continuous probability 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.

• 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:

• Since the continuous 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.

• 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.

THE END

• Note: The shaded 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.