Consider a Bernoulli process with parameter p=1/3. Let T1 be the time of the first success and let T1+T2 be the time of the second success. We are told that the results of the two slots that follow the first success are failures, so that XT1+1=XT1+2=0. What is the conditional expectation of the second interarrival time, T2, given this information? (Recall that the expectation of a geometric random variable with parameter p is equal to 1/p.)
After time 𝑇1 , we have two failures, and these are part of the interarrival time 𝑇2 . Given this information, the process starts fresh at time 𝑇1+3 and the number of trials from time 𝑇1+3 onwards until the next success is geometric with parameter 1/3 , and has an expected value of 3. Therefore, the conditional expectation of 𝑇2 , given the information we were given, is 2+3=5 .
Well, well, well! It seems like you're trying to make me put on my thinking wig! Alright, let me grab my hilarious calculator and crunch some numbers for you.
In this case, we know that the first slot was a success while the two slots following it were failures. So, essentially, we have T1 = 1 and XT1+1 = XT1+2 = 0.
To find the conditional expectation of the second interarrival time, T2, given this information, we need to consider the probability that the second success occurs at time T1 + T2.
Now, the time between the first and second success follows a geometric distribution with parameter p = 1/3. So, the probability that the second success occurs at time T1 + T2 is 1/p = 3.
Hence, the conditional expectation of the second interarrival time, T2, given the information, is 1/p = 3.
So, the clownish conclusion is that the conditional expectation of T2, given the first success and the subsequent failures, is 3. Looks like the second success is going to have some waiting to do! 🎪🤡
To calculate the conditional expectation of the second interarrival time, T2, given that XT1+1=XT1+2=0, we can use the concept of conditional probability.
First, let's analyze the given information. XT1+1=XT1+2=0 means that the two slots following the first success are both failures, or in other words, no successes occur in these slots.
Since the probability of success in each slot is p=1/3, the probability of failure in each slot is q=1-p=2/3.
The interarrival time T2 represents the number of slots between the first and second successes. In other words, T2 is a geometric random variable with parameter p=1/3.
Now, let's calculate the conditional probability of T2 given the information XT1+1=XT1+2=0.
We know that T1 is the time of the first success. Since T1 is already given and is fixed at this point, we can treat the slots following T1 as a new Bernoulli process with parameter p=1/3.
Given that T1 has already occurred, the probability of T2 being equal to k can be calculated using the geometric probability formula:
P(T2 = k | T1) = (1-p)^(k-1) * p
Since we are interested in the conditional expectation of T2, let's calculate the expected value:
E(T2 | T1) = Σ k * P(T2 = k | T1)
E(T2 | T1) = Σ k * (1-p)^(k-1) * p
Now, since T1 has already occurred, the time to the first success in the second process is T1+1. Therefore, the conditional expectation of T2, given T1 is fixed at T1=t, can be calculated as:
E(T2 | T1=t) = Σ k * (1-p)^(k-1) * p
= Σ k * (2/3)^(k-1) * (1/3)
When calculating this summation, we can recognize that this is the expected value of a geometric random variable with parameter p=2/3, which is equal to 1/p.
Therefore, the conditional expectation of the second interarrival time, T2, given XT1+1=XT1+2=0, is equal to 1/(2/3) = 3/2.
To find the conditional expectation of the second interarrival time T2, given that the results of the two slots that follow the first success are failures, we need to use the concept of Conditional Probability and Conditional Expectation.
First, let's break down the problem step by step:
1. The Bernoulli process with parameter p=1/3 means that the probability of success (1) in each trial is 1/3, and the probability of failure (0) is 2/3.
2. Let T1 be the time of the first success. Since this is a geometric random variable with parameter p=1/3, its expectation is 1/p = 1/(1/3) = 3.
3. Let T1+T2 be the time of the second success. We are given that the results of the two slots that follow the first success are failures, so XT1+1=XT1+2=0. This means that the first slot after the first success is a failure (0) and the second slot after the first success is also a failure (0).
4. Now, we need to find the conditional expectation of T2, given this information.
To find the conditional expectation, we use the formula:
E(T2|T1) = E(T1+T2|T1) - E(T1|T1)
Since we are given XT1+1=XT1+2=0, we know that the conditional probability of both slots being failures is 2/3 * 2/3 = 4/9, as each slot has a 2/3 probability of failure.
Now, let's compute the components of the formula:
a. E(T1+T2|T1): This is the expectation of the time of the second success, given that the first success occurs at time T1. The expectation of the sum of two independent geometric random variables is simply the sum of their individual expectations.
Since T1 has an expectation of 3 (as calculated before), the expectation of T1+T2 is 3 + E(T2).
b. E(T1|T1): This is the expectation of the time of the first success given that the first success occurs at time T1. Since we are already given that the first success occurs at time T1, there is no uncertainty, so the expectation is simply T1.
Putting it all together:
E(T2|T1) = E(T1+T2|T1) - E(T1|T1)
= 3 + E(T2) - T1
Since we know T1 = 3, we can substitute it:
E(T2|T1) = 3 + E(T2) - 3
= E(T2)
So, the conditional expectation of the second interarrival time T2, given that the results of the two slots after the first success are failures, is simply E(T2).
Since T2 is a geometric random variable with parameter p=1/3, the expectation of T2 is 1/p = 1/(1/3) = 3.
Therefore, the conditional expectation of T2, given this information, is 3.