All ships travel at the same speed through a wide canal. Each ship takes t 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 λE ships per day. Westbound ships (i.e., ships traveling west) arrive as an independent Poisson process with an arrival rate of λW 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 λE,λW,x,t,v and/or k. Enter 'LE' for λE and 'LW' for λW, 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?
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2) Determine the PDF, fX(x), of the remaining time, X, until the pointer changes direction.
For x≥0, fX(x)=
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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?
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4) Starting at an arbitrary time, we monitor the cross-section of the canal at some fixed location along its length. Let V 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 V.
For v≥0, fV(v)=
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5) What is the probability that the next ship to arrive causes a change in the direction of the pointer?
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6) If we begin monitoring a fixed cross-section of the canal at an arbitrary time, determine the probability mass function pK(k) for K, the total number of ships we observe up to and including the seventh eastbound ship we see. The answer will be of the form pK(k)=(a6)⋅b, for suitable algebraic expressions in place of a and b.
a=
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b=
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1) The probability that the next ship to pass will be westbound can be calculated using the rate of westbound ships and the total arrival rate.
Let λT = λE + λW be the total arrival rate of ships.
The probability that the next ship is westbound can be calculated as the ratio of the rate of westbound ships to the total arrival rate:
P(next ship is westbound) = λW / λT
Answer: λW / (λE + λW)
2) To determine the PDF of the remaining time until the pointer changes direction, we can consider the exponential distribution.
Let T be the time until the pointer changes direction. Since it is a Poisson process, the number of ships passing before the pointer changes direction follows a Poisson distribution with parameter λT.
The time until the pointer changes direction can be considered as the waiting time for the kth ship to pass, where k is the number of ships already passed.
The PDF of the waiting time for the kth ship to pass is given by:
fX(x) = (λT * t)^k * exp(-λT * t) / k!
Answer: (λT * t)^k * exp(-λT * t) / k!
3) The probability that an eastbound ship does not pass any westbound ships during its journey can be calculated as the probability that there are no westbound ships passing during the time it takes for the eastbound ship to travel through the canal.
Let TEB be the time taken for an eastbound ship to traverse the canal.
The number of westbound ships passing in time TEB follows a Poisson distribution with parameter λW * TEB.
The probability of no westbound ship passing during TEB is given by:
P(no westbound ship passes during TEB) = exp(-λW * TEB)
Answer: exp(-λW * TEB)
4) The PDF of V, the amount of time that will have elapsed by the time we observe the seventh eastbound ship, can be calculated using the exponential distribution.
Let VE be the time taken for one eastbound ship to traverse the canal.
The PDF of V is given by the sum of independent exponential random variables (representing the time between the arrivals of each ship), with a rate of (λE + λW).
Since we are interested in the seventh eastbound ship, we need to find the distribution of the sum of each pair of ship arrivals up to the seventh eastbound ship.
Using convolution, the PDF of the sum of two exponential random variables with rates r1 and r2 is given by:
fV(v) = (r1 * r2) / (r2 - r1) * (exp(-r1 * v) - exp(-r2 * v))
Substituting r1 = (λE + λW) and r2 = 7 * (λE + λW), we can calculate the PDF of V.
Answer: (λE + λW) * (7 * (λE + λW)) / (7 * (λE + λW) - (λE + λW)) * (exp(-(λE + λW) * v) - exp(-7 * (λE + λW) * v))
5) To calculate the probability that the next ship to arrive causes a change in the direction of the pointer, we need to consider the arrival rates of the ships.
Let λA be the arrival rate at which ships are arriving.
If the next ship to arrive causes a change in direction, it means that either an eastbound ship arrives while the pointer is pointing west or a westbound ship arrives while the pointer is pointing east.
The probability of an eastbound ship arriving while the pointer is pointing west is given by:
P(eastbound ship arrives while pointer is pointing west) = λE / λA
Similarly, the probability of a westbound ship arriving while the pointer is pointing east is given by:
P(westbound ship arrives while pointer is pointing east) = λW / λA
The overall probability that the next ship to arrive causes a change in the direction of the pointer is the sum of these two probabilities:
P(next ship causes a change) = λE / λA + λW / λA
Answer: (λE + λW) / λA
6) To determine the probability mass function pK(k) for K, the total number of ships observed up to and including the seventh eastbound ship, we can use the binomial distribution.
Let NE be the number of eastbound ships observed up to and including the seventh eastbound ship.
The probability of observing NE eastbound ships out of k total ships, assuming a success probability of λE / (λE + λW) for each ship, is given by the binomial distribution:
pK(k) = C(k, NE) * (λE / (λE + λW))^NE * (λW / (λE + λW))^(k-NE)
Here, C(k, NE) represents the number of combinations of k items taken NE at a time.
Answer:
a = C(k, NE)
b = (λE / (λE + λW))^NE * (λW / (λE + λW))^(k-NE)
To answer these questions, we need to recall some properties of Poisson processes and the definitions of the probabilities and probability mass functions involved. Let's start by reviewing the concepts:
1. Poisson process: A Poisson process is a mathematical model used to describe the occurrence of events over time, where events happen independently and at a constant rate. It has two key parameters: the arrival rate (λ) and the unit of time (t).
2. Probability: The probability of an event happening is a measure of the likelihood of that event occurring. It ranges from 0 (impossible) to 1 (certain). In this context, we will calculate probabilities of events related to ship arrivals and the direction of the pointer.
3. PDF (Probability Density Function): The PDF describes the likelihood of a continuous random variable taking on a specific value. It represents the continuous equivalent of a probability mass function (PMF).
4. PMF (Probability Mass Function): The PMF describes the likelihood of a discrete random variable taking on a specific value. It gives the probability distribution of a discrete random variable.
Now, let's go through each part of the problem together to find the answers:
1) We need to find the probability that the next ship to pass will be westbound (given that the pointer is currently pointing west).
To solve this, we need to consider that the pointer is pointing west, which means the last ship to pass was going westbound. The probability that the next ship will be westbound is equal to the rate of arrival of westbound ships divided by the total arrival rate:
Probability of next ship being westbound = λW / (λE + λW)
2) We need to determine the PDF of the remaining time (X) until the pointer changes direction.
To find the PDF of X, we need to consider that X represents the time until the next ship arrives and causes the pointer to change direction. Since ship arrivals follow a Poisson process, the remaining time until the pointer changes direction also follows an exponential distribution. The PDF of X is given by:
fX(x) = λE * exp(-λE * x) for x ≥ 0
3) We need to find the probability that an eastbound ship does not pass by any westbound ships during its eastward journey through the canal.
This probability is equivalent to the probability that there are no westbound ships arriving during the time it takes for an eastbound ship to traverse the canal (which is t, the time taken by ships to traverse the canal). Since ship arrivals are Poisson processes, the number of westbound ships arriving in time t follows a Poisson distribution with rate λW. The probability of no westbound ships in time t is given by:
Probability of no westbound ships = exp(-λW * t)
4) We need to determine the PDF of V, the amount of time that will have elapsed by the time we observe the seventh eastbound ship.
To calculate the PDF of V, we need to consider that we're counting the amount of time it takes to observe the seventh eastbound ship. Since ship arrivals are Poisson processes, the waiting time until observing the kth arrival (for k ≥ 1) follows a gamma distribution with parameters k and λE. In this case, we're interested in the timing of the seventh eastbound ship, so we can use the gamma distribution with k = 7. The PDF of V is given by:
fV(v) = (λE^7 * v^6 * exp(-λE * v)) / 6! for v ≥ 0
5) We need to find the probability that the next ship to arrive causes a change in the direction of the pointer.
To solve this, we need to consider that the next ship will cause a change in direction if and only if it is traveling in the opposite direction of the pointer. This means the ship must be traveling west if the pointer is currently pointing east, and vice versa. Therefore, the probability of a change in direction is equal to the rate of arrival of ships opposite to the current direction divided by the total arrival rate:
Probability of change in direction = min(λE, λW) / (λE + λW)
6) We need to determine the probability mass function pK(k) for K, the total number of ships observed up to and including the seventh eastbound ship.
To calculate pK(k), we need to consider that the total number of ships observed up to and including the seventh eastbound ship follows a negative binomial distribution. The negative binomial distribution represents the number of successes (eastbound ships) needed before observing a fixed number of failures (westbound ships). In this case, we want to find the probability of observing exactly k-1 failures (westbound ships) and 7 successes (eastbound ships). The PMF of K is given by:
pK(k) = (k-1)C6 * (1/2)^(k-1) * (1/2)^7
where (k-1)C6 represents the binomial coefficient (also known as combinations), which calculates the number of ways to choose 6 failures out of k-1 observations.
Now that we've explained the concepts and how to solve each part of the problem, you can use the provided expressions to calculate the specific answers based on the given values of λE, λW, x, t, v, and k.