1. ## Re: Probability density function

Originally Posted by feli3105
You're right! Wouldn't you mind explaining to me what's the memoryless property and when do I have to use it? It looks like a conditional probability though (because of the "|") but I'm not sure what does $P[L>s+t|L>s]=P[L>t]$ mean.

You're saving my life romsek, thank you so much!! If you ever need help with biology stuff I'd like to help you out.
now you're getting into the territory where it's time to crack your textbook on the exponential distribution and read

2. ## Re: Probability density function

Originally Posted by romsek
now you're getting into the territory where it's time to crack your textbook on the exponential distribution and read
Here are my calculations:

$$P(X<25|X>5)=\frac{P(5<X<25)}{P(X>5)}$$

$$P(5<X<25)=\int_{5}^{25} \lambda e^{-\lambda t} dt=-e^{-\lambda t}\bigg|_{5}^{25}= e^{-\lambda t}\bigg|_{25}^{5}=[1-e^{-25/16}]-[1-e^{-5/16}] \approx 0.52$$

$$P(X<5)=\int_{0}^{5} \lambda e^{-\lambda t} dt=-e^{-\lambda t}\bigg|_{0}^{5}= e^{-\lambda t}\bigg|_{5}^{0}=1-e^{-5/16} \approx 0.27$$

$$P(X>5)=1-P(X<5)=1-0.27=0.73$$

$$P(X<25|X>5)=\frac{0.52}{0.73}\approx 0.71$$

Which has the same probability that $P(X<20)\approx 0.71$. This means, in my opinion, that the peacemaker experiences no wearing out; the probability of a failure before 20 years is the same as the probability of a failure before 25 years. It's counterintuitive though because you would think that any device experiences wearing out during its lifespan. My book says the Weibull distribution is used to model devices that experience wearing out.

3. ## Re: Probability density function

Originally Posted by feli3105
It's counterintuitive though because you would think that any device experiences wearing out during its lifespan. My book says the Weibull distribution is used to model devices that experience wearing out.
Right. The exponential distribution is probably a lousy model for the lifespan of a device for this reason.

Hence the use of the Weibull distribution.

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