# Gamma distribution problem

#### r0bHadz

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 $f(x) = \dfrac{(λ^α)(x^{(α-1)})(e^{(-λx)}) }{Γ(α)}$
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.

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#### StoneTemplePython

Science Advisor
Gold Member
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

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

• WWGD

#### r0bHadz

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

Science Advisor
Gold Member
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

#### Mark44

Mentor
Thread has been moved...

#### StoneTemplePython

Science Advisor
Gold Member
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?

#### r0bHadz

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

Science Advisor
Gold Member
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

Science Advisor
Homework Helper
Dearly Missed
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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#### r0bHadz

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

Science Advisor
Gold Member
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.

#### r0bHadz

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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