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Libniz
A production line yields two types of devices: Type1 devices occur with probability p1 and work for an average time T1, Type2 devices occur with probability p2 and work for an average time T2, such that p1+p2=1 and T2>T1. The time a device works is modeled as a geometrical distribution. Let X be the time a device chosen uniformly at random works. Find the distribution of X.
I am having trouble setting it up
ok i have to think geometric distribution is \(P(n) = p(1-p)^n\) right?
jesus i have no idea. it tell you average is \(T_1\) and i think that means \(\frac{1-p_1}{p_1}=T_1\)
let me try to write this correctly \[T_1=\frac{p_2}{p_1}\] and \[T_2=\frac{p_1}{p_2}\] but i am not sure that helps solve the problem do you have an example to work off of ?
no, this is a practice exam
that's ok, thanks for your help
ok forget that last noise i wrote, it is ridiculous and i am embarrassed maybe it is much simpler we pick device \(T_1\) with probability \(p_1\) and device \(T_2\) with probability \(p_2\) could it be as simple as \(p_1T_1+p_2T_2\)
no that isn't right either, at least i don't think so what text are you using?
I am actually having troubling understanding what is being asked
i think you are being asked the following: pick a device at random, put \(X\) = amount of time it runs. Find the distribution of \(X\)
in other words, find a formula for \(P(X=k)\) the probability the device runs for time \(k\)
but really i am stuck so i should shut up. but perhaps i have a text with something similar, which is why i asked what text you were using
probability and schocastic process by yates
do you know how to combine two geometric distributions?
that is what they are asking
i should delete my embarrassing answer now we will get a real one i hope
can you show me how to combine distributions
No answer from me, I never learned this stuff. But it might be a negative binomial distribution.
if you get the answer, please post it.
I will ask zarkon, when he comes on; hopefully before my quiz though
is that the full problem?
I'm a little confused ... Type1 devices occur with probability p1 ... Type2 devices occur with probability p2 vs "a device chosen uniformly" are they chosen uniformly or with probabilities p1,p2
screen shot in case I mistyped something
I don't think the problem is worded very well especally near the end. here is how I see the problem let \(d1,d2\) be device 1 and device 2 then \[P(X\le x)=P((X\le x,d1)\text{ or }(X\le x,d2))\] \[=P(X\le x|d1)P(d1)+P(X\le x|d2)P(d2)\]
\[=P(X\le x|d1)p1+P(X\le x|d2)p2\]
though I could be misinterpreting the problem (since I don't think it is totally clear)
yeah, I had hard time even understanding what they were asking for
I would ask your prof...problem isn't clear to me and I have taught courses in mathematical statistics.
so this is conditional probability problem?
based on what read it looks like you need to use conditional prob for at least part of the problem