Thus, the mode occurs at x a. In particular, the median occurs at x a.

## Logistic Distribution

We can also derive the moments using the moment generating function, but the calculation is slightly more involved. The standard deviation of the logistic distribution, , is given by:. The logistic mean or MTTF is actually one of the parameters of the distribution, usually denoted as.

In particular, show that f is symmetric about x 0. Here , ,. In particular, show that.

Recall that p 1 p are the odds in favor of an event with probability p. Jump to: Linked 5. Sign up or log in Sign up using Google.

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The lifetime of a mechanical valve is known to follow a logistic distribution.

## The Logistic Distribution

Therefore, also is changing from to. Show that the moment generating function can be written in terms of the beta function and in terms of the gamma function , as follows: Thus, the mode occurs at x 0. Thus, the logistic distribution has the interesting property that the quantiles are the logarithms of the corresponding odds ratios.

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