# Pdf And Cdf Of A Continuous Random Variable

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*In probability theory , a probability density function PDF , or density of a continuous random variable , is a function whose value at any given sample or point in the sample space the set of possible values taken by the random variable can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values , as opposed to taking on any one value.*

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Chapter 2: Basic Statistical Background. Generate Reference Book: File may be more up-to-date. This section provides a brief elementary introduction to the most common and fundamental statistical equations and definitions used in reliability engineering and life data analysis. In general, most problems in reliability engineering deal with quantitative measures, such as the time-to-failure of a component, or qualitative measures, such as whether a component is defective or non-defective.

Our component can be found failed at any time after time 0 e. In this reference, we will deal almost exclusively with continuous random variables. In judging a component to be defective or non-defective, only two outcomes are possible. In this case, the variable is said to be a discrete random variable. The probability density function pdf and cumulative distribution function cdf are two of the most important statistical functions in reliability and are very closely related.

When these functions are known, almost any other reliability measure of interest can be derived or obtained. We will now take a closer look at these functions and how they relate to other reliability measures, such as the reliability function and failure rate. The pdf and cdf give a complete description of the probability distribution of a random variable. The following figure illustrates a pdf. The pdf represents the relative frequency of failure times as a function of time.

The cdf represents the cumulative values of the pdf. That is, the value of a point on the curve of the cdf represents the area under the curve to the left of that point on the pdf. The total area under the pdf is always equal to 1, or mathematically:. The well-known normal or Gaussian distribution is an example of a probability density function. The pdf for this distribution is given by:. Again, this is a 2-parameter distribution. Since this function defines the probability of failure by a certain time, we could consider this the unreliability function.

Subtracting this probability from 1 will give us the reliability function, one of the most important functions in life data analysis.

The reliability function gives the probability of success of a unit undertaking a mission of a given time duration. The following figure illustrates this. This is the same as the cdf. Reliability and unreliability are the only two events being considered and they are mutually exclusive; hence, the sum of these probabilities is equal to unity.

Conditional reliability is the probability of successfully completing another mission following the successful completion of a previous mission. The time of the previous mission and the time for the mission to be undertaken must be taken into account for conditional reliability calculations.

The conditional reliability function is given by:. The failure rate function enables the determination of the number of failures occurring per unit time. Omitting the derivation, the failure rate is mathematically given as:. This gives the instantaneous failure rate, also known as the hazard function. It is useful in characterizing the failure behavior of a component, determining maintenance crew allocation, planning for spares provisioning, etc.

Failure rate is denoted as failures per unit time. The mean life function, which provides a measure of the average time of operation to failure, is given by:. The MTTF, even though an index of reliability performance, does not give any information on the failure distribution of the component in question when dealing with most lifetime distributions.

Because vastly different distributions can have identical means, it is unwise to use the MTTF as the sole measure of the reliability of a component. It represents the centroid of the distribution. For individual data, the median is the midpoint value. A statistical distribution is fully described by its pdf. In the previous sections, we used the definition of the pdf to show how all other functions most commonly used in reliability engineering and life data analysis can be derived.

Different distributions exist, such as the normal Gaussian , exponential, Weibull, etc. In fact, there are certain references that are devoted exclusively to different types of statistical distributions. These distributions were formulated by statisticians, mathematicians and engineers to mathematically model or represent certain behavior.

For example, the Weibull distribution was formulated by Waloddi Weibull and thus it bears his name. Some distributions tend to better represent life data and are most commonly called "lifetime distributions".

A more detailed introduction to this topic is presented in Life Distributions. Navigation menu Personal tools Log in. Namespaces Page Discussion. Views Read View source View history. This page was last edited on 1 August , at Creative Commons Attribution.

Index Chapter 2. Basic Statistical Background.

## Probability density function

Say you were to take a coin from your pocket and toss it into the air. While it flips through space, what could you possibly say about its future? Will it land heads up? More than that, how long will it remain in the air? How many times will it bounce?

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Math Statistics and probability Random variables Continuous random variables. Probability density functions. Probabilities from density curves. Practice: Probability in density curves.

Exploratory Data Analysis 1. EDA Techniques 1. Probability Distributions 1. Probability distributions are typically defined in terms of the probability density function. However, there are a number of probability functions used in applications. For a continuous function, the probability density function pdf is the probability that the variate has the value x.

Let X be a continuous random variable with pdf f and cdf F.

## Cumulative distribution function

Chapter 2: Basic Statistical Background. Generate Reference Book: File may be more up-to-date. This section provides a brief elementary introduction to the most common and fundamental statistical equations and definitions used in reliability engineering and life data analysis.

*Metrics details. In this paper a comprehensive survey of the different methods of generating discrete probability distributions as analogues of continuous probability distributions is presented along with their applications in construction of new discrete distributions. The methods are classified based on different criterion of discretization.*

#### 4.1.3 Functions of Continuous Random Variables

В глубине души она понимала, что абсурдно обвинять в нерадивости Стратмора, который был беззаветно предан своему делу и воспринимал все зло мира как свое личное. Попрыгунчик был любимым детищем коммандера, смелой попыткой изменить мир. Увы, как и большинство других поисков божества, она закончилась распятием. - Хорошо, - сказала. - Я немного погорячилась. - Немного? - Глаза Бринкерхоффа сузились. - У Стратмора стол ломится от заказов.

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Прямо перед ним, откинувшись на груду старых подушек, лежал пожилой человек с ярко-белой гипсовой повязкой на правом запястье. ГЛАВА 21 Голос американца, звонившего Нуматаке по прямой линии, казался взволнованным: - Мистер Нуматака, в моем распоряжении не больше минуты. - Хорошо. Полагаю, вы получили обе копии ключа.

*За десертом в ночных ресторанах он задавал ей бесконечные вопросы. Где она изучала математику.*

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