Characterizing random processes

[ashah99]

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:

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

 

[FactChecker]

The problem statement is not visible. It's just a jpeg file name.

 

[ashah99]

The problem statement is not visible. It's just a jpeg file name.

Interesting, I edited my posted, hope it shows up

 

[Mark44]

The problem statement is not visible. It's just a jpeg file name.

Interesting, I edited my posted, hope it shows up

It shows up now.

 

[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.

 

[FactChecker]

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.

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. ?

 

[ashah99]

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?

 

[FactChecker]

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

Agree

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

Sum of iid Bernoulli RVs is binomial

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

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

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.

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

Agree

I'm not expert enough to help on the rest.

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

 

[ashah99]

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: