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2 Let {N(t)} be Poisson process with rate A and arrival times S1, S2 , a fixed time &, we are given that N(t) = 2.Suppose that, forFind the conditional pdf for ...

Question

2 Let {N(t)} be Poisson process with rate A and arrival times S1, S2 , a fixed time &, we are given that N(t) = 2.Suppose that, forFind the conditional pdf for the second arrival time given that N(t) = 2. (Hint: Start with the conditional cdf:) Find the conditional expectation of the second arrival time given that N(t) = 2.

2 Let {N(t)} be Poisson process with rate A and arrival times S1, S2 , a fixed time &, we are given that N(t) = 2. Suppose that, for Find the conditional pdf for the second arrival time given that N(t) = 2. (Hint: Start with the conditional cdf:) Find the conditional expectation of the second arrival time given that N(t) = 2.



Answers

Find the indicated probability using the Poisson distribution. Find $P(2)$ when $\mu=1.5$

This is a very basic question off the exponential distribution. Now what happens over here is that we're here. We have been given that Lambda is equal to one an exponential distribution Mu is equal toe one by Lambda is equal to one by Lambda, which means that the mean is going to be equal to that the mean, Mean is going to be equal to one. So I can say that between two consecutive arrivals, the time is going to be one right? What is the question? The question is what is expected time between two successive arrivals. It is going to be one now. What happens in exponential distribution is that Sigma Sigma is equal toe the mean between standard deviation and the mean are equal, which means this answer is also going to be one. So our sigma is also equal to one. Now what we can do is for the next two questions, we can either find the values by hand or we can use statistical Softwares as I'm going toe show you over here. The question is, you have to find probability that X is less than equal to four right p that are X is less than or equal to four. That excess less than or equal to four. Now, what I have is if I go to the stab aware, I already have this calculated. I put my left eye is equal to one. My X is equal to four. And I want less than or equal toe so I can see that my area or I can see that my probability is 0.981 succeed. This is going to be my answer apart. See, we can do this by hand. Definitely. But using the software is always easy. And this is what we're going to be doing even in our real life. Example. So this is zero point 981680.98168 Okay, and then we want to find probability. That too, is OK access between two and five. So my probability that my ex is greater than equal to two and less than equal to five. Now what do I do? Okay, I will find my probability that my ex is less than equal to five and I will subtract the probability that my ex is less than equal to two. So this is what I can do to find this. So if I go to the stab again if I put my X is equal to four. Sorry, Max is equal to for less than equal to. And this thing is Oh, yes. So this is 0.86466 rate. So this thing is zero point 86466 Yeah. Now let us go toe. Faith, we're here. I'm going to have five. And this is going to be 0.993 to 6.993 to six. 0.993 to 6. And if I subtract this, if I use a calculator for this, this thing is 0.99 3 to 6, minus 0.86 466 This is 4660.1286 This is my answer for this question. 0.1286 This is my answer for this question. And this is how we go about doing this question


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