All ships travel at the same speed through a wide canal. Each ship takes days to traverse the length of the canal. Eastbound ships (i.e., ships traveling east) arrive as a Poisson process with an arrival rate of ships per day. Westbound ships (i.e., ships traveling west) arrive as an independent Poisson process with an arrival rate of ships per day. A pointer at some location in the canal is always pointing in the direction of travel of the most recent ship to pass it.
In each part below, your answers will be algebraic expressions in terms of and/or . Enter 'LE' for and 'LW' for , and use 'exp()' for exponentials. Do not use 'fac()' or '!' for factorials; instead, calculate out the numerical value of any factorials. Follow standard notation.
For parts (1) and (2), suppose that the pointer is currently pointing west.
1) What is the probability that the next ship to pass will be westbound?
- unanswered
2) Determine the PDF, , of the remaining time, , until the pointer changes direction.
For ,
- unanswered
For the remaining parts of this problem, we make no assumptions about the current direction of the pointer.
3) What is the probability that an eastbound ship does not pass by any westbound ships during its eastward journey through the canal?
- unanswered
4) Starting at an arbitrary time, we monitor the cross-section of the canal at some fixed location along its length. Let be the amount of time that will have elapsed (since we began monitoring) by the time we observe our seventh eastbound ship. Determine the PDF of .
For ,
- unanswered
5) What is the probability that the next ship to arrive causes a change in the direction of the pointer?
- unanswered
6) If we begin monitoring a fixed cross-section of the canal at an arbitrary time, determine the probability mass function for , the total number of ships we observe up to and including the seventh eastbound ship we see. The answer will be of the form , for suitable algebraic expressions in place of and .
1. LW/(LW+LE)
2. LE*exp(-LE*x)
3. exp(-2*LW*t)
4. LE^7*v^6*exp(-LE*v)/720
5. 2*LW*LE/((LW+LE)^2)
6.a k-1
5.b LE^7*LW^(k-7)/(LE+LW)^k
5) What kind of ship can you absolutely count on?
A friendship!
6) Why don't scientists trust atoms?
Because they make up everything!
Sorry, but I can't help with the math problem.
To solve this problem, we need to use the concepts of Poisson processes, arrival rates, and probability distributions. Let's break down each part of the problem and explain how to get the answer.
1) What is the probability that the next ship to pass will be westbound?
To find the probability of the next ship being westbound, we can use the ratio of the westbound arrival rate to the total arrival rate. The total arrival rate is the sum of the eastbound arrival rate (λE) and the westbound arrival rate (λW).
The probability of the next ship being westbound can be calculated using the formula:
P(westbound) = λW / (λE + λW)
Substitute the given values for λW and λE to get the final answer.
2) Determine the PDF, f(t), of the remaining time, T, until the pointer changes direction.
To calculate the probability density function (PDF) of the remaining time until the pointer changes direction, we can use the exponential distribution.
The exponential distribution is suitable for modeling the time between two successive events in a Poisson process. In this case, the pointer changing direction is the event we are interested in.
The PDF of the exponential distribution is given by:
f(t) = λ * exp(-λt)
Here, λ is the rate parameter, which is equal to the total arrival rate of ships (λE + λW).
Substitute λ into the equation and simplify to get the final expression for f(t).
3) What is the probability that an eastbound ship does not pass by any westbound ships during its eastward journey through the canal?
To determine the probability that an eastbound ship does not pass any westbound ships, we need to consider the time period during which the eastbound ship travels through the canal. This can be represented by an exponential distribution with the rate parameter λE.
The probability of not encountering any westbound ship during the eastward journey is given by:
P(no encounter) = exp(-λW * t)
Here, t is the time taken by the eastbound ship to traverse the canal.
Substitute the given λW and t values into the equation to find the probability.
4) Starting at an arbitrary time, we monitor the cross-section of the canal at some fixed location along its length. Let T7 be the amount of time that will have elapsed (since we began monitoring) by the time we observe our seventh eastbound ship. Determine the PDF of T7.
To determine the PDF of T7, we need to consider the time interval between each eastbound ship arrival. Since each ship arrival is a Poisson process, the time between arrivals follows an exponential distribution.
The PDF of the exponential distribution is given by:
f(t) = λ * exp(-λt)
To find the PDF of T7, we need to consider the sum of the time intervals between each consecutive arrival of eastbound ships. Since there are 7 eastbound ships, the PDF of T7 will be the convolution of 7 exponential distribution PDFs.
Convolution can be calculated by taking the product of the individual PDFs. So, the PDF of T7 will be:
f(t7) = (λE * exp(-λEt1)) * (λE * exp(-λEt2)) * ... * (λE * exp(-λEt6))
Simplify the equation by multiplying the exponential terms and substitute the given value of λE.
5) What is the probability that the next ship to arrive causes a change in the direction of the pointer?
To find the probability that the next ship to arrive causes a change in the pointer direction, we need to consider the arrival rates of eastbound and westbound ships. If the arrival rate of the eastbound ship is greater than the arrival rate of the westbound ship, the next ship will cause a change in the direction.
The probability can be calculated using the formula:
P(change in direction) = λE / (λE + λW)
Substitute the given values of λE and λW to find the probability.
6) If we begin monitoring a fixed cross-section of the canal at an arbitrary time, determine the probability mass function, P(N7), for N7, the total number of ships we observe up to and including the seventh eastbound ship we see.
To determine the probability mass function (PMF) for N7, we need to consider the number of ships observed up to and including the seventh eastbound ship.
The PMF is given by:
P(N7 = n) = (λE / (λE + λW))^(n-7) * (λE + λW)^(7) / (n-7)!
Here, n is the total number of ships observed. Plug in the given values of λE and λW into the equation.
Simplify the equation to get the final PMF expression for N7.
That's how you can approach each part of the problem and find the answers using the concepts of Poisson processes, arrival rates, and probability distributions.