A distribution function, also known as a cumulative distribution function (CDF), is a mathematical function that describes the probability that a random variable takes on a certain value or falls within a certain range of values. It gives the cumulative probability of all values of the random variable up to a certain point.
A random variable is a variable whose value is determined by the outcome of a random process. It can take on a range of possible values, each with a certain probability of occurring. Random variables are used to model real-world phenomena that involve uncertainty or randomness.
A distribution function is the mathematical representation of a probability distribution. It maps all possible values of a random variable to their corresponding probabilities. The area under the distribution function curve between two values represents the probability of the random variable falling within that range.
There are many types of probability distributions, but some of the most commonly used include the normal distribution, binomial distribution, Poisson distribution, and exponential distribution. Each distribution has its own unique shape and characteristics, and is used to model different types of random processes.
Distribution functions are essential in statistics because they allow us to calculate probabilities of events and make predictions based on data. They are used to analyze and interpret data, make inferences about populations, and test hypotheses. They are also used to generate random numbers for simulations and other statistical analyses.