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- 1.3 – The Discrete Probability Density Function
- 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
- Probability distribution

## 1.3 – The Discrete Probability Density Function

In probability and statistics, a randomvariable is a variable whose value is subject to variations due to chance i. As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value even if unknown ; rather, it can take on a set of possible different values, each with an associated probability.

Random variables can be classified as either discrete that is, taking any of a specified list of exact values or as continuous taking any numerical value in an interval or collection of intervals. The mathematical function describing the possible values of a random variable and their associated probabilities is known as a probability distribution.

Discrete random variables can take on either a finite or at most a countably infinite set of discrete values for example, the integers. Their probability distribution is given by a probability mass function which directly maps each value of the random variable to a probability. Discrete Probability Disrtibution : This shows the probability mass function of a discrete probability distribution. A set not containing any of these points has probability zero.

Examples of discrete random variables include the values obtained from rolling a die and the grades received on a test out of Continuous random variables, on the other hand, take on values that vary continuously within one or more real intervals, and have a cumulative distribution function CDF that is absolutely continuous.

As a result, the random variable has an uncountable infinite number of possible values, all of which have probability 0, though ranges of such values can have nonzero probability. The resulting probability distribution of the random variable can be described by a probability density, where the probability is found by taking the area under the curve.

As notated on the figure, the probabilities of intervals of values corresponds to the area under the curve. Selecting random numbers between 0 and 1 are examples of continuous random variables because there are an infinite number of possibilities.

Probability distributions for discrete random variables can be displayed as a formula, in a table, or in a graph. A discrete probability distribution can be described by a table, by a formula, or by a graph. Notice that these two representations are equivalent, and that this can be represented graphically as in the probability histogram below.

Probability Histogram : This histogram displays the probabilities of each of the three discrete random variables. The formula, table, and probability histogram satisfy the following necessary conditions of discrete probability distributions:.

Sometimes, the discrete probability distribution is referred to as the probability mass function pmf. The probability mass function has the same purpose as the probability histogram, and displays specific probabilities for each discrete random variable. The only difference is how it looks graphically. Probability Mass Function : This shows the graph of a probability mass function. All the values of this function must be non-negative and sum up to 1. Discrete Probability Distribution : This table shows the values of the discrete random variable can take on and their corresponding probabilities.

The expected value of a random variable is the weighted average of all possible values that this random variable can take on. In probability theory, the expected value or expectation, mathematical expectation, EV, mean, or first moment of a random variable is the weighted average of all possible values that this random variable can take on.

The weights used in computing this average are probabilities in the case of a discrete random variable. The expected value may be intuitively understood by the law of large numbers: the expected value, when it exists, is almost surely the limit of the sample mean as sample size grows to infinity. More informally, it can be interpreted as the long-run average of the results of many independent repetitions of an experiment e.

This is intuitive: the expected value of a random variable is the average of all values it can take; thus the expected value is what one expects to happen on average. Average Dice Value Against Number of Rolls : An illustration of the convergence of sequence averages of rolls of a die to the expected value of 3. Privacy Policy. Skip to main content. Probability and Variability.

Search for:. Discrete Random Variables. Learning Objectives Contrast discrete and continuous variables. Key Takeaways Key Points A random variable is a variable taking on numerical values determined by the outcome of a random phenomenon. A discrete random variable has a countable number of possible values. The probability of each value of a discrete random variable is between 0 and 1, and the sum of all the probabilities is equal to 1.

A continuous random variable takes on all the values in some interval of numbers. A density curve describes the probability distribution of a continuous random variable, and the probability of a range of events is found by taking the area under the curve.

Key Terms random variable : a quantity whose value is random and to which a probability distribution is assigned, such as the possible outcome of a roll of a die discrete random variable : obtained by counting values for which there are no in-between values, such as the integers 0, 1, 2, ….

Probability Distributions for Discrete Random Variables Probability distributions for discrete random variables can be displayed as a formula, in a table, or in a graph.

Learning Objectives Give examples of discrete random variables. Key Terms discrete random variable : obtained by counting values for which there are no in-between values, such as the integers 0, 1, 2, …. Expected Values of Discrete Random Variables The expected value of a random variable is the weighted average of all possible values that this random variable can take on.

Learning Objectives Calculate the expected value of a discrete random variable. Licenses and Attributions. CC licensed content, Shared previously.

## 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable

In probability theory and statistics , a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. For instance, if X is used to denote the outcome of a coin toss "the experiment" , then the probability distribution of X would take the value 0. Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey , etc. A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. To define probability distributions for the specific case of random variables so the sample space can be seen as a numeric set , it is common to distinguish between discrete and continuous random variables.

Discrete and Continuous Random Variables:. A variable is a quantity whose value changes. A discrete variable is a variable whose value is obtained by counting. A continuous variable is a variable whose value is obtained by measuring. A random variable is a variable whose value is a numerical outcome of a random phenomenon.

There are two types of random variables , discrete random variables and continuous random variables. The values of a discrete random variable are countable, which means the values are obtained by counting. All random variables we discussed in previous examples are discrete random variables. We counted the number of red balls, the number of heads, or the number of female children to get the corresponding random variable values. The values of a continuous random variable are uncountable, which means the values are not obtained by counting. Instead, they are obtained by measuring.

discrete variables and distributions. Page 4. 4. Probability Distributions for Continuous Variables. Suppose the random variable and is denoted by Z. Its pdf is.

## Probability distribution

Previous: 1. Next: 1. Usually we are interested in experiments where there is more than one outcome, each having a possibly different probability.

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. 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 probability mass function, denoted by f x.

In probability and statistics, a randomvariable is a variable whose value is subject to variations due to chance i. As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value even if unknown ; rather, it can take on a set of possible different values, each with an associated probability. Random variables can be classified as either discrete that is, taking any of a specified list of exact values or as continuous taking any numerical value in an interval or collection of intervals.

When introducing the topic of random variables, we noted that the two types — discrete and continuous — require different approaches. The equivalent quantity for a continuous random variable, not surprisingly, involves an integral rather than a sum. Several of the points made when the mean was introduced for discrete random variables apply to the case of continuous random variables, with appropriate modification.

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Где-то в самом низу шахты воспламенились процессоры. ГЛАВА 105 Огненный шар, рвущийся наверх сквозь миллионы силиконовых чипов, производил ни на что не похожий звук. Треск лесного пожара, вой торнадо, шипение горячего гейзера… все они слились в гуле дрожащего корпуса машины. Это было дыхание дьявола, ищущее выхода и вырывающееся из закрытой пещеры. Стратмор так и остался стоять на коленях, парализованный ужасающим, неуклонно приближающимся звуком.

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*У них было много общего: настойчивость, увлеченность своим делом, ум. Иногда ей казалось, что Стратмор без нее пропадет; ее любовь к криптографии помогала коммандеру отвлечься от завихрений политики, напоминая о молодости, отданной взламыванию шифров.*