Basic Statistical Background

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.

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.

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

See Also

See Also

Notes

Notes

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

See Also

See Also

Notes

Notes