Characterizing random processes

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SUMMARY

This discussion focuses on characterizing various random processes, specifically using distributions such as Poisson, Bernoulli, Geometric, Exponential, Negative Binomial, Binomial, and Erlang. The user presents their attempts at identifying the correct distributions for a given problem, including Poisson(3) for raindrop counts and Exp(3) for arrival rates. Key insights include the clarification that the output of a rain gauge follows a Poisson distribution and that the sum of independent Poisson random variables is also Poisson. The user seeks confirmation and guidance on their understanding of parameters for these distributions.

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  • Understanding of Poisson distribution and its properties
  • Familiarity with Bernoulli and Binomial distributions
  • Knowledge of Geometric and Exponential distributions
  • Concept of Negative Binomial distribution and its parameters
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ashah99
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Homework Statement
Please see below.
Relevant Equations
Distribution parameters
Hello. I would like to kindly request some help with a multi-part problem on identifying random processes as an intro topic from my stats course. I’m fairly uncertain with this topic so I suspect my attempt is mostly incorrect, especially when specifying the parameters, and I would be grateful for any guidance and help with understanding this better.

The problem statement:

rv_prob.jpg


My attempt:
a) Poisson( ##\lambda ## = 3)

b) Poisson( ##\lambda ## = 9), where ## X_1, X_2, X_3 ## are all ~Poisson(3), and I believe a property of Poission RVs is that a sum is also Poisson with ##\lambda ## = ##\lambda_1 + \lambda_2 + \lambda_3##

Sum of iid Bernoulli RVs is binomial

c) Bernoulli(##e^{-1}##)

d) Geometric(p=1/2), not sure on the parameter
Waiting times between successive flashes are iid geo RVs where the rate is 1 per sec

e) Exp(3) Since the rate is 3 arrivals/sec

f) NegBinomial(p=1/2, k=3) not sure on the parameter
waiting time for k-th arrival, where k = 3

g) Binomial(n = 10, p = 1/2) not sure on the parameter

h) Erlang(##\lambda ## = 3, k = 3) not sure on the parameter
Waiting time for continuous Poisson process where lambda is the arrival rate and k=3
 
Last edited:
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The problem statement is not visible. It's just a jpeg file name.
 
FactChecker said:
The problem statement is not visible. It's just a jpeg file name.
Interesting, I edited my posted, hope it shows up
 
FactChecker said:
The problem statement is not visible. It's just a jpeg file name.

ashah99 said:
Interesting, I edited my posted, hope it shows up
It shows up now.
 
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Likes   Reactions: FactChecker and ashah99
I’m quite thrown off with how the rain gauge outputting a random sequence at a rate of one every second comes into play in this problem, especially when specifying the partners for the RVs.
 
ashah99 said:
I’m quite thrown off with how the rain gauge outputting a random sequence at a rate of one every second comes into play in this problem,
The output of the rain gauge is the Poisson RV. The number of raindrops hitting the rain gauge in one second has the Poisson distribution, Pois(3). It is related to a lot of other distributions.
ashah99 said:
especially when specifying the partners for the RVs.
I don't know what "partners" means. Are you talking about the related distributions like exponential, etc. ?
 
Last edited:
FactChecker said:
The output of the rain gauge is the Poisson RV. The number of raindrops hitting the rain gauge in one second has the Poisson distribution, Pois(3). It is related to a lot of other distributions.

I don't know what "partners" means. Are you talking about the related distributions like exponential, etc. ?
Sorry, I meant parameters (i.e. the lambda in Poisson(3) ). I'm quite sure of my answers for parts a and b but the rest is making me confused. Would you be willing to check and guide through mistakes?
 
ashah99 said:
My attempt:
a) Poisson( ##\lambda ## = 3)
Agree
ashah99 said:
b) Poisson( ##\lambda ## = 9), where ## X_1, X_2, X_3 ## are all ~Poisson(3), and I believe a property of Poission RVs is that a sum is also Poisson with ##\lambda ## = ##\lambda_1 + \lambda_2 + \lambda_3##
Agree
ashah99 said:
Sum of iid Bernoulli RVs is binomial

c) Bernoulli(##e^{-1}##)
Shouldn't the parameter be ##Pois(X=0, \lambda = 3) = \lambda^0 e^{-\lambda}/0!##
ashah99 said:
d) Geometric(p=1/2), not sure on the parameter
Waiting times between successive flashes are iid geo RVs where the rate is 1 per sec
This seems like a negative binomial to me.
ashah99 said:
e) Exp(3) Since the rate is 3 arrivals/sec
Agree

I'm not expert enough to help on the rest.
ashah99 said:
f) NegBinomial(p=1/2, k=3) not sure on the parameter
waiting time for k-th arrival, where k = 3

g) Binomial(n = 10, p = 1/2) not sure on the parameter

h) Erlang(##\lambda ## = 3, k = 3) not sure on the parameter
Waiting time for continuous Poisson process where lambda is the arrival rate and k=3
 
FactChecker said:
Agree

Agree

Shouldn't the parameter be ##Pois(X=0, \lambda = 3) = \lambda^0 e^{-\lambda}/0!##

This seems like a negative binomial to me.

Agree

I'm not expert enough to help on the rest.
For part (c) I think you are right, I think I mixed it up with the one output per second output from the rain gauge. So, I believe the final answer should be ##Bernoulli(e^{-3})##.

So, for part (d), you think it is NegBin, but I'm not not sure what the p and k parameters should be (see notes below)?

So, for the rest of the parts, I used these notes of discrete vs. continuous Poisson that I found online, so my latter answers are based off those:
1670769931569.png
 

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