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.
0.0036
Continue from and use the same setup as above.
Find the expected value of T.
To find the expected value of T, we need to calculate the probability of each possible value of T and then multiply it by that value.
In this scenario, T can take on the values 1, 2, 3, 4, and so on.
Let's calculate the probability of each possible value of T:
P(T=1) = probability of having an arrival in time slot 10 = (1/3) * (2/3)^2 = 4/27
P(T=2) = probability of having no arrival in time slot 10 and having an arrival in either time slot 8 or 9 = (2/3) * (1/3) * (2/3) = 4/27
P(T=3) = probability of having no arrival in time slot 10, having an arrival in both time slot 8 and 9, and having no arrival in time slot 7 or earlier = (2/3) * (2/3) * (2/3) * (1/3)^3 = 8/243
P(T=4) = probability of having no arrival in time slot 10, having an arrival in both time slot 8 and 9, and having an arrival in time slot 7 or earlier = (2/3) * (2/3) * (2/3) * (2/3) * (1/3)^4 = 16/729
To calculate the expected value of T, we multiply each value of T by its probability and sum them up:
E(T) = (1 * P(T=1)) + (2 * P(T=2)) + (3 * P(T=3)) + (4 * P(T=4)) + ...
E(T) = (1 * 4/27) + (2 * 4/27) + (3 * 8/243) + (4 * 16/729) + ...
To calculate the expected value, we can sum up these terms:
E(T) = (4/27) + (8/27) + (24/243) + (64/729) + ...
Using the formula for the sum of an infinite geometric series, we have:
E(T) = (4/27) * (1/(1 - (2/3)))
Simplifying further:
E(T) = (4/27) * (3/1) = 12/27 = 4/9
Therefore, the expected value of T is 4/9.
To find the expected value of T, we need to calculate the average number of time slots until the next arrival.
Given that the arrival probability at each time slot is p=1/3, this implies that the probability of no arrival (denoted by q) at each time slot is q = 1 - p = 1 - 1/3 = 2/3.
Let's consider the different possibilities for T:
1. T = 1: This means that the next arrival occurs immediately after the observer arrives at time slot 10. The probability of this happening is q * q = (2/3) * (2/3) = 4/9.
2. T = 2: This means that there is one time slot between the observer's arrival at time slot 10 and the next arrival at time slot 11. The probability of this happening is q * p * q = (2/3) * (1/3) * (2/3) = 4/27.
3. T = 3: This means that there are two time slots between the observer's arrival at time slot 10 and the next arrival at time slot 12. The probability of this happening is q * p * p * q = (2/3) * (1/3) * (1/3) * (2/3) = 8/81.
4. T = 4: This means that there are three time slots between the observer's arrival at time slot 10 and the next arrival at time slot 13. The probability of this happening is q * p * p * p * q = (2/3) * (1/3) * (1/3) * (1/3) * (2/3) = 16/243.
To calculate the expected value of T, we multiply each possibility by its respective outcome (T value) and sum them up:
Expected value of T = (1 * (4/9)) + (2 * (4/27)) + (3 * (8/81)) + (4 * (16/243))
Calculating this expression, we find that the expected value of T is approximately 2.302.
To find the expected value of T, we can consider the possible values of T and their respective probabilities.
Let's consider the possible values of T:
1. T = 1: This means that the next arrival occurs immediately after the time slot of the last arrival. The probability of this happening is p * (1-p) = (1/3) * (2/3) = 2/9.
2. T = 2: This means that there is exactly one time slot between the time slot of the last arrival and the time slot of the next arrival. The probability of this happening is (1-p) * p * (1-p) + p * (1-p) * (1-p) = (2/3) * (1/3) * (2/3) + (1/3) * (2/3) * (2/3) = 8/27.
3. T = 3: This means that there are exactly two time slots between the time slot of the last arrival and the time slot of the next arrival. The probability of this happening is (1-p) * (1-p) * p * (1-p) + p * (1-p) * (1-p) * (1-p) = (2/3) * (2/3) * (1/3) * (2/3) + (1/3) * (2/3) * (2/3) * (2/3) = 16/81.
4. T = 4: This means that there are exactly three time slots between the time slot of the last arrival and the time slot of the next arrival. The probability of this happening is (1-p) * (1-p) * (1-p) * p * (1-p) + p * (1-p) * (1-p) * (1-p) * (1-p) = (2/3) * (2/3) * (2/3) * (1/3) * (2/3) + (1/3) * (2/3) * (2/3) * (2/3) * (2/3) = 64/243.
To find the expected value of T, we multiply each value of T by its respective probability and sum them up:
E(T) = 1 * (2/9) + 2 * (8/27) + 3 * (16/81) + 4 * (64/243) = 2/9 + 16/27 + 48/81 + 256/243 = (2 + 64/27 + 16/27 + 256/243) = 66/27 + 16/27 + 256/243 = 338/243 = 1.392.
Therefore, the expected value of T is 1.392.