Finding probability related to Poisson and Exponential Distribution

  • #1
songoku
2,314
330
Homework Statement
Please see below
Relevant Equations
Poisson Distribution

Exponential Distribution
1699953819487.png


My attempt:
(i) ##\lambda =3##

(ii)
(a) ##P(N_{2} \geq 1=1-P(N_{2} =0)=1-e^{-6} \frac{(-6)^0}{0!}=0.997##

(b) ##P(N_{4} \geq 3)=1-P(N_{4} \leq 2)=0.999##

(c) ##P(N_{1} \geq 2) = 1-P(N_{4} \leq 1)=0.8##

Do I even understand the question correctly for part (i) and (ii)?(iii) The expectation of exponential distribution is ##\frac{1}{\lambda}## so the answer is ##\frac{1}{3}##?

Thanks
 
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  • #2
songoku said:
Do I even understand the question correctly for part (i) and (ii)?
Yes, I think so, but there are a couple of errors in notation.
In (a), (-6)0 should be 60.
In (b), N4 should be N1.
Otherwise it looks correct.
For (iii), what are you calculating the expectation of?
 
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  • #3
Here is a simple sanity check of your answer for (ii)(a):
On average, there are three failures per unit time. Your answer for part (ii)(a) is that it is almost certain that there are no failures in two time units. Does that seem right?
 
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  • #4
D'oh! should have seen that!
 
  • #5
FactChecker said:
Here is a simple sanity check of your answer for (ii)(a):
On average, there are three failures per unit time. Your answer for part (ii)(a) is that it is almost certain that there are no failures in two time units. Does that seem right?
Yeah, it does not make sense.

mjc123 said:
there are a couple of errors in notation.

In (b), N4 should be N1.
Sorry I don't understand why it should be N1, maybe I am misunderstanding something again.

N = number of failures
t = time interval

Is that correct?Revised attempt:
(ii)
(a) ##P(N_{2} = 0)=e^{-6}=0.0025##

(b) still the same as post #1

(c) still the same as post #1, with a revision ##N_{4}## should be ##N_{1}##
Oh wait, maybe you mean the error in notation is for (c) instead of (b)? @mjc123

mjc123 said:
For (iii), what are you calculating the expectation of?
Oh my god, the expectation of Sn

$$E(S_{n}-S_{n-1})=\frac{1}{\lambda}$$
$$E(S_{n})-E(S_{n-1})=\frac{1}{\lambda}$$
$$E(S_{n})=\frac{1}{\lambda}+E(S_{n-1})$$

Not even sure this is the correct approach

Thanks
 
  • #6
Oh no, yes I meant c not b. I must have been half asleep that day. Apologies for confusion.
(iii), you're on the right track.
S0 = 0
E(S1 - S0) = 1/λ
E(S1) = 1/λ
E(S2 - S1) = 1/λ
E(S2) = ?
etc.
 
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  • #7
I understand. Thank you very much for the help mjc123 and FactChecker
 

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