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.
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.
1 answer
3 answers