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Gamma distribution problem

  • Thread starter r0bHadz
  • Start date
180
15
Problem Statement
A certain system is based on two independent modules, A and B. A failure of any module causes a failure of the whole system. The lifetime of each module has a Gamma distribution, with parameters α λ given in the table: [COMPONENT A: α:3 λ:1] [COMPONENT B: α:2 λ:2]

a)What is the probability that the system works at least 2 years without a failure? (I already have the answer for this, which is .06195, estimated to .062)

b)Given that the system failed during the first 2 years, what is the probability that it failed due to the failure of component B (but not component A)?
Relevant Equations
gamma density [itex]f(x) = \dfrac{(λ^α)(x^{(α-1)})(e^{(-λx)}) }{Γ(α)}[/itex]
I'm lost. First one was easy to calculate, second one is harder.

I have:

P{a fails before 2 yrs} = .323325
P{b fails before 2 yrds} = .90844

P{system doesnt fail for 2 years or longer} = .062
P{system does fail before 2 years} = .938

P{A and B fail before 2 yrs} = .29372
P{before 2 years A fails but B doesn't} = .0296
P{before 2 years A doesnt fail but B fails} = .6147

so in answering B, I assume it is asking me:

P{System failed due to B and not A | system failed during first 2 yrs} =

(.6147*.938)/.938

which = .6147 but this is not the right answer. I don't know where my logic is wrong.
 
Last edited:

StoneTemplePython

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I'm lost. First one was easy to calculate, second one is harder...
Can we move this to the calculus forums and fix the LaTeX?

Gamma's.. aren't pre-calc.

As for the LaTeX, you are using ##\propto## instead of, I'm guessing, ##\alpha## which is quite confusing.

As far as your bottom statement

##\frac{( P{before 2 years A doesnt fail but B fails} = .6147 * P{system does fail before 2 years} = .938 ) }{P{system does fail before 2 years} = .938}##
try wrapping it in "\text{}" it read as

##\frac{( P{\text{ before 2 years A doesnt fail but B fails}} = .6147 * P{\text{ system does fail before 2 years}} = .938 ) }{P{\text{ system does fail before 2 years}} = .938}##
 
180
15
Can we move this to the calculus forums and fix the LaTeX?

Gamma's.. aren't pre-calc.

As for the LaTeX, you are using ##\propto## instead of, I'm guessing, ##\alpha## which is quite confusing.

As far as your bottom statement

##\frac{( P{before 2 years A doesnt fail but B fails} = .6147 * P{system does fail before 2 years} = .938 ) }{P{system does fail before 2 years} = .938}##
try wrapping it in "\text{}" it read as

##\frac{( P{\text{ before 2 years A doesnt fail but B fails}} = .6147 * P{\text{ system does fail before 2 years}} = .938 ) }{P{\text{ system does fail before 2 years}} = .938}##

Sorry most of my threads from this course have been in this forum so I thought it would fit.

And yes, I do mean to use 'alpha' but I had trouble finding it :/
 

StoneTemplePython

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Sorry most of my threads from this course have been in this forum so I thought it would fit.

And yes, I do mean to use 'alpha' but I had trouble finding it :/
put a \ before "alpha" i.e. in LaTeX use ##\alpha## to get alpha... what you use was ##\propto## i.e. the symbol for proportional to, which is quie different.

You can right click my LaTeX and do "show math as -> Tex Commands" to see the underlying LaTeX
 
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Thread has been moved...
 

StoneTemplePython

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What background knowledge do you have on the gamma distribution? Do you know about Poisson processes?

We're dealing with positive integer values for ##\alpha## so this is that important special case called an Erlang Distribution. (Erlang comes up in queuing problems and is very natural in that if you look at a Poisson process and the density for the time until kth arrival, it is Erlang of order k...)

Supposing you understand some of the above:
Do you understand that component ##A## is the sum of ##X_1 + X_2 + X_3## and component ##B## is the sum of ##Y_1 + Y_2## where ##X_i## are iid exponential random variables with parameter ##\lambda = 1## and the ##Y_i## are iid exponential random variables with parameter ##\lambda = 2##?

If so, how can you simplify the problem and use Poisson process "stuff" to get a nice easy answer?

Supposing none of that made sense:
what do you know about Gamma's and what technique has been suggested in class / in your text?
 
180
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My book never mentions Erlang. I understand the third paragraph.

What I understand about Gamma distributions:

When a procedure consists of α amount of steps and each step takes an exponential amount of time λ, then the total time has a gamma distribution.

What I think I did wrong in this problem is, my logic is wrong somewhere for the conditional probability
 

StoneTemplePython

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What I understand about Gamma distributions:

When a procedure consists of α amount of steps and each step takes an exponential amount of time λ, then the total time has a gamma distribution.
Let's use this. I'd say ##\alpha ## "arrivals" not "steps" here, but I understand you. In terms of a Poisson process -- do you understand how a Poisson process can be interpreted as counting "arrivals" where each one is iid exponential with parameter ##\lambda##? If so, how can you use that to tackle the problem?

Less satisfying but equivalent approach -- do you know how to get (look up?) the CDF for a Gamma distribution?

your question (a) is equivalent to
##Pr\{A \gt 2\} \cdot Pr\{B\gt 2\}##
because they are independent. How can you get this with use the use of the CDFs?

your question (b) as a first cut I'd suggest using the complement of what you compute in part (a)... i.e. 1- answer for part (a) is total probability of this happening. (This ends up in the denominator -- apply bayes rule). Can you work out in sets what this complement refers to?

then using conditioning and integration, compute the probability that ##B## and ##A## both fail before 2, but ##B## fails first. This is the tricky part. The other, easier part is ##Pr\{A \gt 2\} \cdot Pr\{B\leq 2\}## -- these two are mutually exclusive events so their probabilities add.

There should be more clever approaches... and again some poisson process ideas comes to mind.
- - - -
I am finding it rather tough to figure out how to help you here as your relevant equation is just a definition of the gamma density -- it doesn't suggest much about what you know and don't know. And your work shown is just numbers -- it isn't clear what techniques you used to arrive at them or what techniques are being pushed in your text...
 
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Ray Vickson

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My book never mentions Erlang. I understand the third paragraph.

What I understand about Gamma distributions:

When a procedure consists of α amount of steps and each step takes an exponential amount of time λ, then the total time has a gamma distribution.

What I think I did wrong in this problem is, my logic is wrong somewhere for the conditional probability
An Erlang distribution is just a Gamma distribution with an integer value of ##\alpha##. It is true that an Erlang random variable is the sum of ##\alpha## iid exponential (##\lambda##) random variable; usually these are referred to as stages or phases, not steps. (I have also not heard of them as being called "arrivals" when they are merely part of an a-priori Erlang, but of course they are arrivals when we refer to such things as the time of the nth arrival in a Poisson process, for example.)

Your characterization of a general Gamma (with non-integer ##\alpha##) is incorrect: if ##\alpha = 3.46##, how can you have 3.46 stages or arrivals or whatever you want to call them?

There is something your question does not mention: what happens after a failure? Is the component immediately replaced by an identical, but brand-new component? Or, do you just let the system sit in failed mode for the whole future? This could make a difference to probability calculations.

(1) No replacement. For ##T=2## (yrs) the probability that A fails in ##(0,T)## but B does not fail in ##(0,T)## is
$$P(\text{A fails but not B}) = P(T_A < T, T_B > T) = \int_0^T f_A(t) \, dt \int_T^\infty f_B(t) \, dt.$$

(2) Replacement after failure. The probability that A fails once and B does not fail is
$$P(\text{A once but not B}) = \sum_t P(T_{A1} = t < T, T_{A2}>T-t, T_B > T) \\
= \int_0^T \left[ f_A(t) \int_t^T f_A(s) \, ds\right] \, dt \int_T^\infty f_B(t) \, dt$$
 
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180
15
Hmm I seem to have gotten the right answer now, which is .6147/.938

Ray I don't think I wrote α=3.46 anywhere mate??
 

StoneTemplePython

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Hmm I seem to have gotten the right answer now, which is .6147/.938

Ray I don't think I wrote α=3.46 anywhere mate??
Do you mind posting your solution -- it would be a bit more satisfying for us to see the result written up.

What he's saying here with ##\alpha = 3.46## is an example... Erlangs use natural numbers for ##\alpha## and your intuition breaks for "steps" if ##\alpha## is not a natural number. So the gamma is a bit more general. The fact that squaring a standard normal gives a gamma with parameters 1/2, 1/2 seems relevant though probably a bit too far afield.
 
180
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I answered this question:"
b)Given that the system failed during the first 2 years, what is the probability that it
failed due to the failure of component B (but not component A)? "

Using:
P{before 2 years A doesnt fail but B fails} = .6147

divided by:
P{system does fail before 2 years} = .938

Gives the solution, which is .6553

To calculate the first two, I used the gamma density formula on the "relevant equations" tab in the OP, with the values given in the OP for the parameters
 

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