# Distributions associated with Bathtub Curve

#### leaning

##### Involved In Discussions
Hello!

I'm trying to make one sheet to take the place of most of my notes. This diagram combines many of the pictures and concepts I've found into one slide, but I need you folk's help.

The areas of the curve are defined by various distributions. I listed them at the top in red, but can't find anything to show what section of the curve goes with which distribution. I found that the green line is from the exponential distribution, but what about the yellow and brown and blue curves? So, names without homes. I can't find any diagram to this detail that helps with this.

If anyone sees anything else wrong or that could be added, please let me know.

All the best,
leaning

#### Attachments

• bathtub curve.pdf
636.7 KB · Views: 133

#### Miner

##### Forum Moderator
The blue bathtub curve arises from superimposing the brown, yellow and green curves upon each other.

The most common distributions have already been identified for you in red at the top of the chart. The decreasing failure rate is the brown curve, the constant failure rate is the green line, and the increasing failure rate is the yellow line.

The brown curve may also be described using the loglogistic distribution, The yellow curve may also be described by the logistic, loglogistic, and smallest extreme value distributions.

#### leaning

##### Involved In Discussions
Miner,

Thanks for your reply. I knew about what the blue,yellow, green, and brown curves were represented AFA failures, just not what distributions go with each. AFA the red text above the areas, I put those there. I want to break this down better still and color the red text to match the curve that goes with it. So is it like this?

Green (constant failure): exponential
Brown (decreasing failure): lognormal or Weibull B<1 or loglogistic
Yellow (increasing failure): normal or lognormal or Weibull >1 or logistic, loglogistic, and smallest extreme value distributions.

But then I have a "Weibull B=1" left over not associated with a curve.

??

Regards,
leaning

#### Miner

##### Forum Moderator
Green (constant failure): exponential, Weibull B=1
Brown (decreasing failure): lognormal or Weibull B<1 or loglogistic
Yellow (increasing failure): normal or lognormal or Weibull >1 or logistic, loglogistic, and smallest extreme value distributions.

Now it is correct.

Note: The exponential distribution equation results when you take the Weibull distribution equation, set B=1 and simplify the equation.

#### leaning

##### Involved In Discussions
Miner,

Perfect! Thanks for sticking with me through this.

For this forum, would you rather we remove the Version 1 attachment and then replace it with a Version 2, or keep the Version 1 on here, and attach the Version 2, 3,4 with it?

Regards,
leaning

#### Miner

##### Forum Moderator
Just attach the later versions in your subsequent posts. That way, viewers can see the progression throughout the thread.

#### leaning

##### Involved In Discussions
Miner,

Here's v2.

Thanks for your feedback. Let me know if y'all see anything else.

Regards,
leaning

#### Attachments

• Bathtub Curve v2.pdf
571.4 KB · Views: 130

#### leaning

##### Involved In Discussions
Miner,

One other thing I want to add. If look at the attached, it has formulas for the various sections. I know that B is for the constant section. Which one is for the 1) decreasing and 2) increasing failure sections?

?

Regards,
leaning

#### Attachments

• Choices.jpg
9.7 KB · Views: 219

#### normzone

Trusted Information Resource
That's a very helpful document, my thanks to all of you.

#### Miner

##### Forum Moderator
One other thing I want to add. If look at the attached, it has formulas for the various sections. I know that B is for the constant section. Which one is for the 1) decreasing and 2) increasing failure sections?

A is incorrect. It is a product of a failure rate and time and would result in the number of failures, not a reliability.
B is incorrect. Lambda is the failure rate, not the reliability.
C is incorrect. It is the product of a failure rate and a constant reliability.
D is the correct formula for constant failure rate

Lambda is only applicable to an assumed exponential distribution and would never be used in a formula for an increasing or decreasing hazard rate. The correct formula would depend on the equation for the assumed distribution.

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