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In the first case, where the random variable can take on only two discrete values (let's say ''defective =0'' and ''non-defective=1''>), the variable is said to be a ''discrete random variable''. In the second case, our product can be found failed at any time after time ''0'', i.e. at ''12.4 hours'' or at ''100.12 miles'' and so forth, thus ''X'' can take on any value in this range. In this case, our random variable ''X'' is said to be a ''continuous random variable''.
In the first case, where the random variable can take on only two discrete values (let's say ''defective =0'' and ''non-defective=1''>), the variable is said to be a ''discrete random variable''. In the second case, our product can be found failed at any time after time ''0'', i.e. at ''12.4 hours'' or at ''100.12 miles'' and so forth, thus ''X'' can take on any value in this range. In this case, our random variable ''X'' is said to be a ''continuous random variable''.


==='''The Probability Density and Cumulative Distribution Functions'''===
{{ProbabilityDensitynCumulativeDistributionFunctions}}
{{ProbabilityDensitynCumulativeDistributionFunctions}}



Revision as of 22:16, 3 January 2012

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Chapter 2: Basic Statistical Background


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Chapter 2  
Basic Statistical Background  

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Available Software:
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More Resources:
Weibull++ Examples Collection


Statistical Background

In 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.


Random Variables

In general, most problems in reliability engineering deal with quantitative measures, such as the time-to-failure of a component, or whether the component fails or does not fail. In judging a component to be defective or non-defective, only two outcomes are possible. We can then denote a random variable X as representative of these possible outcomes (i.e. defective or non-defective). In this case, X is a random variable that can only take on these values.

In the case of times-to-failure, our random variable X can take on the time-to-failure (or time to an event of interest) of the product or component and can be in a range from 0 to infinity (since we do not know the exact time a priori).

In the first case, where the random variable can take on only two discrete values (let's say defective =0 and non-defective=1>), the variable is said to be a discrete random variable. In the second case, our product can be found failed at any time after time 0, i.e. at 12.4 hours or at 100.12 miles and so forth, thus X can take on any value in this range. In this case, our random variable X is said to be a continuous random variable.

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis

Chapter 2: Basic Statistical Background


Weibullbox.png

Chapter 2  
Basic Statistical Background  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection


Statistical Background

In 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.


Random Variables

In general, most problems in reliability engineering deal with quantitative measures, such as the time-to-failure of a component, or whether the component fails or does not fail. In judging a component to be defective or non-defective, only two outcomes are possible. We can then denote a random variable X as representative of these possible outcomes (i.e. defective or non-defective). In this case, X is a random variable that can only take on these values.

In the case of times-to-failure, our random variable X can take on the time-to-failure (or time to an event of interest) of the product or component and can be in a range from 0 to infinity (since we do not know the exact time a priori).

In the first case, where the random variable can take on only two discrete values (let's say defective =0 and non-defective=1>), the variable is said to be a discrete random variable. In the second case, our product can be found failed at any time after time 0, i.e. at 12.4 hours or at 100.12 miles and so forth, thus X can take on any value in this range. In this case, our random variable X is said to be a continuous random variable.

Template loop detected: Template:ProbabilityDensitynCumulativeDistributionFunctions

The Reliability Function


The Conditional Reliability Function

Template:ConditionalReliabilityFunction

The Failure Rate Function

Template:FailureRateFunction

The Mean Life Function

The mean life function, which provides a measure of the average time of operation to failure, is given by:

[math]\displaystyle{ \mu = m =\int_{0,\gamma}^{\infty}t\cdot f(t)dt }[/math]

It should be noted that this is the expected or average time-to-failure and is denoted as the MTBF (Mean-Time-Before Failure) and is also called MTTF (Mean-Time-To-Failure) by many authors.

The Median Life Function

Median life,[math]\displaystyle{ \breve{T} }[/math] is the value of the random variable that has exactly one-half of the area under the [math]\displaystyle{ pdf }[/math] to its left and one-half to its right. The median is obtained from:

[math]\displaystyle{ \int_{-\infty}^{\breve{T}}f(t)dt=0.5 }[/math]

For sample data, e.g. 12, 20, 21, the median is the midpoint value, or 20 in this case.

The Mode Function

The modal (or mode) life, is the maximum value of [math]\displaystyle{ T }[/math] that satisfies:

[math]\displaystyle{ \frac{d\left[ f(t) \right]}{dt}=0 }[/math]

For a continuous distribution, the mode is that value of the variate which corresponds to the maximum probability density (the value where the [math]\displaystyle{ pdf }[/math] has its maximum value).

Additional Resources

References

See Also

See Also

Notes

Notes

The Reliability Function


The Conditional Reliability Function

Template:ConditionalReliabilityFunction

The Failure Rate Function

Template:FailureRateFunction

The Mean Life Function

The mean life function, which provides a measure of the average time of operation to failure, is given by:

[math]\displaystyle{ \mu = m =\int_{0,\gamma}^{\infty}t\cdot f(t)dt }[/math]

It should be noted that this is the expected or average time-to-failure and is denoted as the MTBF (Mean-Time-Before Failure) and is also called MTTF (Mean-Time-To-Failure) by many authors.

The Median Life Function

Median life,[math]\displaystyle{ \breve{T} }[/math] is the value of the random variable that has exactly one-half of the area under the [math]\displaystyle{ pdf }[/math] to its left and one-half to its right. The median is obtained from:

[math]\displaystyle{ \int_{-\infty}^{\breve{T}}f(t)dt=0.5 }[/math]

For sample data, e.g. 12, 20, 21, the median is the midpoint value, or 20 in this case.

The Mode Function

The modal (or mode) life, is the maximum value of [math]\displaystyle{ T }[/math] that satisfies:

[math]\displaystyle{ \frac{d\left[ f(t) \right]}{dt}=0 }[/math]

For a continuous distribution, the mode is that value of the variate which corresponds to the maximum probability density (the value where the [math]\displaystyle{ pdf }[/math] has its maximum value).

Additional Resources

References

See Also

See Also

Notes

Notes