## Libniz 3 years ago 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.

1. Libniz

@satellite73

2. Libniz

I am having trouble setting it up

3. satellite73

ok i have to think geometric distribution is $$P(n) = p(1-p)^n$$ right?

4. Libniz

yes

5. satellite73

jesus i have no idea. it tell you average is $$T_1$$ and i think that means $$\frac{1-p_1}{p_1}=T_1$$

6. satellite73

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 ?

7. Libniz

no, this is a practice exam

8. Libniz

that's ok, thanks for your help

9. satellite73

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 $$p_1T_1+p_2T_2$$

10. satellite73

no that isn't right either, at least i don't think so what text are you using?

11. Libniz

I am actually having troubling understanding what is being asked

12. satellite73

i think you are being asked the following: pick a device at random, put $$X$$ = amount of time it runs. Find the distribution of $$X$$

13. satellite73

in other words, find a formula for $$P(X=k)$$ the probability the device runs for time $$k$$

14. satellite73

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

15. Libniz

probability and schocastic process by yates

16. satellite73

nope, sorry

17. Libniz

@phi

18. phi

do you know how to combine two geometric distributions?

19. Libniz

no

20. phi

that is what they are asking

21. satellite73

i should delete my embarrassing answer now we will get a real one i hope

22. Libniz

can you show me how to combine distributions

23. phi

No answer from me, I never learned this stuff. But it might be a negative binomial distribution.

24. Libniz

alright, thanks anyway

25. phi

26. Libniz

I will ask zarkon, when he comes on; hopefully before my quiz though

27. Libniz

@zarkon

28. Zarkon

is that the full problem?

29. Libniz

yes

30. Zarkon

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

31. Libniz

screen shot in case I mistyped something

32. Zarkon

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 $P(X\le x)=P((X\le x,d1)\text{ or }(X\le x,d2))$ $=P(X\le x|d1)P(d1)+P(X\le x|d2)P(d2)$

33. Zarkon

$=P(X\le x|d1)p1+P(X\le x|d2)p2$

34. Zarkon

though I could be misinterpreting the problem (since I don't think it is totally clear)

35. Libniz

yeah, I had hard time even understanding what they were asking for

36. Zarkon

I would ask your prof...problem isn't clear to me and I have taught courses in mathematical statistics.

37. Libniz

so this is conditional probability problem?

38. Zarkon

based on what read it looks like you need to use conditional prob for at least part of the problem