Characterizing random processes

In summary: Continuous Poisson:-Has an infinite number of discrete samples (e.g. timepoints within a day)-Each sample has the same probability-Parameters are determined by the rate at which the event occurs.Discrete Poisson:-Has a finite number of samples (e.g. timepoints within a day)-Each sample has a different probability-Parameters are determined by the number of samples.
  • #1
ashah99
60
2
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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  • #2
The problem statement is not visible. It's just a jpeg file name.
 
  • #3
FactChecker said:
The problem statement is not visible. It's just a jpeg file name.
Interesting, I edited my posted, hope it shows up
 
  • #4
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 FactChecker and ashah99
  • #5
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.
 
  • #6
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:
  • #7
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?
 
  • #8
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
 
  • #9
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
 

1. What is a random process?

A random process is a sequence of events or outcomes that occur in a random or unpredictable manner. It is a mathematical model used to describe systems or phenomena that exhibit randomness, such as the behavior of particles, the stock market, or the weather.

2. How do you characterize a random process?

Characterizing a random process involves identifying and quantifying its statistical properties, such as its mean, variance, and correlation. This can be done through mathematical models, simulations, or data analysis techniques.

3. What is the difference between a stationary and non-stationary random process?

A stationary random process is one in which the statistical properties (e.g. mean, variance) remain constant over time. In contrast, a non-stationary random process has statistical properties that change over time, making it more unpredictable.

4. How do you model a random process?

There are several ways to model a random process, depending on the specific application and the available data. Some common techniques include Markov chains, autoregressive models, and stochastic differential equations.

5. What are some real-world examples of random processes?

Random processes can be found in many natural and man-made systems. Examples include the movement of molecules in a gas, the flow of traffic on a highway, the fluctuation of stock prices, and the occurrence of earthquakes. They are also used in fields such as finance, physics, biology, and engineering to model and understand complex systems.

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