# Construction of a probability distribution for a random variable

There are two major classes of probability distributions. Discrete Continuous A discrete random variable has a finite or countable number of possible values. A continuous random variable takes on an uncountably infinite number of possible values e. It is frequently used to represent binary experiments, such as a coin toss. Random variables and probability distributions A random variable is a numerical description of the outcome of a statistical experiment. A random variable that may assume only a finite number or an infinite sequence of values is said to be discrete; one that may assume any value in some interval on the real number line is said to be continuous.

For instance, a random variable representing the number of automobiles sold at a particular dealership on one day would be discrete, while a random variable representing the weight of a person in kilograms or pounds would be continuous.

## Discrete random variables

The probability distribution for a random variable describes how the probabilities are distributed over the values of the random variable. For a discrete random variable, x, the probability distribution is defined by a Construction of a probability distribution for a random variable mass function, denoted by f x.

This function provides the probability for each value of the random variable. In the development of the probability function for a discrete random variable, two conditions must be satisfied: A continuous random variable may assume any value in an interval on the real number line or in a collection of intervals.

Since there is an infinite number of values in any interval, it is not meaningful to talk about the probability that the random variable will take on a specific value; instead, the probability that a continuous random variable will lie within a given interval is considered.

In the continuous case, the counterpart of the probability mass function is the probability density functionalso denoted by f x. For a continuous random variable, the probability density function provides the height or value of the function at any particular value of x; it does not directly give the probability of the random variable taking on a specific value.

However, the area under the graph of f x corresponding to some interval, obtained by computing the integral of f x over that interval, provides the probability that the variable will take on a value within that interval. A probability density function must satisfy two requirements: In the discrete case the weights are given by the probability mass function, and in the continuous case the weights are given by the probability density function.

The formulas for computing the expected values of discrete and continuous random variables are given by equations 2 and 3, respectively. The formulas for computing the variances of discrete and continuous random variables are given by equations 4 and 5, respectively. Since the standard deviation is measured in the same units as the random variable and the variance is measured in squared units, the standard deviation is often the preferred measure.

The binomial probability mass function equation 6 provides the probability that x successes will occur in n trials of a binomial experiment.

A binomial experiment has four properties: The Poisson distribution The Poisson probability distribution is often used as a model of the number of arrivals at a facility within a given period of time.

For instance, a random variable might be defined as the number of telephone calls coming into an airline reservation system during a period of 15 minutes. If the mean number of arrivals during a minute interval is known, the Poisson probability mass function given by equation 7 can be used to compute the probability of x arrivals.

For example, suppose that the mean number of calls arriving in a minute period is The normal distribution The most widely used continuous probability distribution in statistics is the normal probability distribution.

Like all normal distribution graphs, it is a bell-shaped curve. Probabilities for the normal probability distribution can be computed using statistical tables for the standard normal probability distribution, which is a normal probability distribution with a mean of zero and a standard deviation of one.

The tables for the standard normal distribution are then used to compute the appropriate probabilities. There are many other discrete and continuous probability distributions. Other widely used discrete distributions include the geometric, the hypergeometric, and the negative binomial; other commonly used continuous distributions include the uniform, exponential, gamma, chi-square, beta, t, and F.

Estimation It is often of interest to learn about the characteristics of a large group of elements such as individuals, households, buildings, products, parts, customers, and so on. All the elements of interest in a particular study form the population. Because of time, cost, and other considerations, data often cannot be collected from every element of the population.

In such cases, a subset of the population, called a sample, is used to provide the data. Data from the sample are then used to develop estimates of the characteristics of the larger population. The process of using a sample to make inferences about a population is called statistical inference.

Characteristics such as the population mean, the population variance, and the population proportion are called parameters of the population. Characteristics of the sample such as the sample mean, the sample variance, and the sample proportion are called sample statistics.

There are two types of estimates: A point estimate is a value of a sample statistic that is used as a single estimate of a population parameter. No statements are made about the quality or precision of a point estimate.Construction of independent random variables.

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## Special probability distributions

The product measure construction allows us to extend Lemma 4 (Creating a random variable with a specified distribution): Exercise 18 (Creation of new. Constructing a probability distribution for random variable.

Practice: Constructing probability distributions. Probability models example: frozen yogurt. Practice: Probability models. Valid discrete probability distribution examples. Probability with discrete random variable example.

Construction of random variables. Ask Question. up vote 0 down vote favorite. That got me thinking that I never actually learned how a random variable with a certain distribution is constructed. Random variable and probability space notions.

Existence of iid random variables. 1. Random Variables. Formally, a random variable is a function that assigns a real number to each outcome in the probability space. Define your own discrete random variable for the uniform probability space on the right and sample to find the empirical distribution.

4 Continuous Random Variables and Probability Distributions (Ed 8) – Chapter 4 - and Cengage. 2 Continuous r.v. A random variable X is continuous if possible values comprise either a single interval on the number line or a distribution in all of probability and statistics.

RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 1. DISCRETE RANDOM VARIABLES Deﬁnition of a Discrete Random Variable. A random variable X is said to be discrete if it can assume only a ﬁnite or countable inﬁnite number of distinct values.

Random Variables