Basic Statistical Definitions

Random Variables

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

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

In the first case in which the random variable can take on 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 hr or at 100 hr 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 continous random variable. In this reference, we will deal almost exclusively with continuous random variables.

The Probability Density and Cumulative Density Functions

Designations

From probability & statistics, given a continuous random variable X, we denote:

The probability density function, pdf, as f(x).
The cumulative density function, cdf, as F(x).

The pdf and cdf give a complete description of the probability distribution of a random variable.

Definitions

If X is a continuous random variable, then the probability density function, pdf, of X is a function f(x) such that for two numbers, a and b with a b:

(1)

That is, the probability that X takes on a value in the interval [a, b] is the area under the density function from a to b.

The cumulative distribution function, cdf, is a function F(x), of a random variable X, and is defined for a number x by:

(2)

That is, for a number x, F(x) is the probability that the observed value of X will be at most x.

Note that depending on the function denoted by f(x), or more specifically the distribution denoted by f(x), the limits will vary depending on the region over which the distribution is defined. For example, for all the life distributions considered in this reference, this range would be [0, + ].

Graphical Representation of the pdf and cdf

Mathematical Relationship Between the pdf and cdf

The mathematical relationship between the pdf and cdf is given by:

(3)

where s is a dummy integration variable.

Conversely:

In plain English, the cdf is the area under the probability density function, up to a value of x, if so chosen. The total area under the pdf is always equal to 1, or mathematically:

An example of a probability density function is the well-known normal distribution, for which the pdf is given by:

where μ is the mean and σ is the standard deviation. The normal distribution is a two parameter distribution, i.e. with two parameters μ and σ.

Another is the lognormal distribution, whose pdf is given by:

where is the mean of the natural logarithms of the times-to-failure and is the standard deviation of the natural logarithms of the times-to-failure. Again, this is a two-parameter distribution.

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