Consider a Bernoulli process, with arrival probability at each time slot equal to p=1/3.
An observer arrives at time slot 10 and sees that no arrival took place in that slot. A passerby informs the observer that there was exactly one arrival during the preceding two time slots (i.e., time slots 8 and 9) but has no additional information about which of the two it was.
Let T be the number of time slots starting from the time slot of the last arrival (before slot 10) and until the time slot where the next arrival comes.
For example, if the last arrival came at time 8 and the next arrival comes at time 13, T is the number of slots in the interval {8,9,10,11,12,13}, so that T=6.
(Enter a numerical expression or a decimal accurate to at least 3 decimals places.)
Find the probability that T=4.
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Problem 4(b)
2.0 points possible (graded, results hidden)
Continue from and use the same setup as above.
Find the expected value of T.
Well, let's calculate the probability that T=4. The observer arrives at time slot 10 and is informed that there was exactly one arrival in the preceding two slots, 8 and 9. Since the arrival probability in each slot is p=1/3, the probability of an arrival in any given slot is 1/3.
To have T=4, there must have been no arrival in slots 11, 12, and 13, and an arrival in slot 14. The probability of no arrival in a single slot is (1-p) = (1-1/3) = 2/3, and the probability of an arrival in a single slot is p = 1/3. Therefore, the probability of T=4 is (2/3)^3 * (1/3) = 8/243, or approximately 0.0329.
Now, let's move on to finding the expected value of T. The expected value of a random variable can be calculated by summing the product of each possible outcome and its respective probability.
Since T represents the number of slots between two arrivals, it can take on values from 1 to infinity. Let's calculate the expected value using the formula for expected value:
E(T) = Σ(x * P(x)), where x denotes each possible value of T and P(x) denotes the probability of T=x.
Since we know that T=4 has a probability of 8/243, we can continue the sum from T=5 onwards.
E(T) = 4 * (8/243) + 5 * (8/243)^2 + 6 * (8/243)^3 + ...
This is an infinite geometric series with first term a=4 and common ratio r=(8/243), because each subsequent term is multiplied by (8/243). Using the formula for the sum of an infinite geometric series, the expected value of T can be calculated as:
E(T) = a / (1 - r) = 4 / (1 - 8/243) ≈ 4.097
Therefore, the expected value of T is approximately 4.097.
To find the expected value of T, we need to consider the probabilities of different values of T occurring.
If T = 1, it means that the next arrival happened immediately after the last arrival at time slot 9. The probability of this happening is (1-p) = (1-1/3) = 2/3.
If T = 2, it means that the next arrival happened one slot after the last arrival at time slot 9. The probability of this happening is p * (1-p) = (1/3) * (2/3) = 2/9.
If T = 3, it means that the next arrival happened two slots after the last arrival at time slot 9. The probability of this happening is (1-p) * (1-p) * p = (2/3) * (2/3) * (1/3) = 4/27.
If T = 4, it means that the next arrival happened three slots after the last arrival at time slot 9. The probability of this happening is (1-p) * (1-p) * (1-p) = (2/3) * (2/3) * (2/3) = 8/27.
From the above probabilities, we can calculate the expected value of T as follows:
Expected value of T = (1 * (2/3)) + (2 * (2/9)) + (3 * (4/27)) + (4 * (8/27)) = 2/3 + 4/9 + 12/27 + 32/27 = 80/27.
Therefore, the expected value of T is 80/27.
To find the probability that T = 4, we can use the concept of conditional probability and the given information.
First, let's analyze the possible scenarios for the last two time slots (slots 8 and 9):
Scenario 1: Arrival in slot 8, no arrival in slot 9.
Scenario 2: No arrival in slot 8, arrival in slot 9.
Scenario 3: No arrival in both slot 8 and slot 9.
Since we know that there was exactly one arrival in the last two time slots, the probability of each scenario occurring is equal.
Now, let's calculate the probabilities for each scenario:
Scenario 1: Probability of arrival in slot 8 * Probability of no arrival in slot 9
= p * (1 - p)
= (1/3) * (1 - 1/3)
= 1/9
Scenario 2: Probability of no arrival in slot 8 * Probability of arrival in slot 9
= (1 - p) * p
= (2/3) * (1/3)
= 2/9
Scenario 3: Probability of no arrival in both slot 8 and slot 9
= (1 - p) * (1 - p)
= (2/3) * (2/3)
= 4/9
Since these scenarios are mutually exclusive, we can calculate the probability of T = 4 as the sum of the probabilities of the first two scenarios:
Probability(T = 4) = Probability of Scenario 1 + Probability of Scenario 2
= 1/9 + 2/9
= 3/9
≈ 0.333
Therefore, the probability that T = 4 is approximately 0.333.
Now, let's move on to finding the expected value of T.
The expected value of T can be calculated by considering all possible values of T and their corresponding probabilities. To find these probabilities, we need to consider all the scenarios for the last two time slots.
Let's denote E[T] as the expected value of T.
For T = 1:
The only scenario where T = 1 is when there is an arrival in slot 10.
Probability(T = 1) = p = 1/3
For T > 1:
The possible scenarios for the last two time slots are Scenario 2 and Scenario 3.
Probability(T = 2) = Probability of Scenario 2 = 2/9
Probability(T = 3) = Probability of Scenario 3 * Probability of no arrival in slot 10 (given there was no arrival in slot 10)
= (4/9) * (2/3)
= 8/27
For T = 4 and beyond:
To calculate the probabilities for T = 4 and beyond, we need to consider the previous probabilities and apply the concept of conditional probability.
Probability(T = 4) = Probability of Scenario 1 + Probability of Scenario 2 = 3/9
Probability(T = 5) = Probability of Scenario 3 * Probability of no arrival in slot 10 * Probability of no arrival in slot 11 (given T = 4)
= (4/9) * (2/3) * (2/3)
= 16/81
We can see that the pattern continues as we consider higher values of T.
To calculate the expected value, we multiply each value of T by its corresponding probability and sum them up:
E[T] = (1/3) * 1 + (2/9) * 2 + (8/27) * 3 + (3/9) * 4 + (16/81) * 5 + ...
However, calculating the exact value of E[T] using this infinite sum can be quite complicated.
One approach to simplify the calculation is to notice that the probabilities form a geometric series with a common ratio of (2/3). This allows us to use the formula for the sum of an infinite geometric series:
E[T] = (1/3) * 1 + (2/9) * 2 + (8/27) * 3 + ... = (1/3) * (1/(1-(2/3))) = (1/3) * (3/2) = 1/2
Therefore, the expected value of T is 1/2.
a) 8/81
b) 3
1. 0.2276
2. 0.3333