Problem 3 3 puntos posibles (calificados, resultados...

Problem 3 3 puntos posibles (calificados, resultados ocultos) Consider a Poisson process with rate $\lambda = 2$ and let $T$ be the time of the first arrival. 1. Find the conditional PDF of $T$ given that the second arrival came before time $t = 1$. Enter an expression in terms of $\lambda$ and $t$. 2. Find the conditional PDF of $T$ given that the third arrival comes exactly at time $t = 1$.

Transcript

00:01 Hello, so to solve these, we can consider the properties of a poisson process with rate here, lambda is equal to 2.

00:07 We can use the fact that the inter -arrival times in a poisson process are independent and exponentially distributed.

00:15 So for part 1, so our setup lambda equals 2, therefore we have that f sub t of t is going to be be equal to 2 times e to the negative 2t, where t is greater than or equal to 0.

00:33 We are conditioning on the event at the arrival time, s sub 2 is less than 1.

00:39 So we recall that s sub 2 is going to be equal to t plus s, where s is the time between the first and second arrivals, and also follows an exponential distribution with parameter lambda equals 2.

00:58 So then calculating the conditional probability, the probability of s sub 2 less than 1 can be computed using the joint density of t and s and integrating then over the appropriate range.

01:10 So this is going to be equal to the integral going from 0 to 1 and then the integral going from 0 to 1 minus t of 4 times e to the negative 2t times e to the negative 2s.

01:25 And then we're integrating here ds so solving this we do the inner integral um first right so the integral from 0 to 1 of 2 times e to negative 2t minus 2 e to negative 2 dt that is going to give us um 1 minus e to the negative 2 minus 2 e to the negative 2 which is going to be just equal uh to 1 minus 3 times e to the negative 2 and then to find the conditional pdf of t, we plug into our formula and where we get that, this is going to be equal to 2 times e to the negative 2t times 1 minus e to the negative 2 times 1 minus t, all over 1 minus 3 times e to the negative 2t...

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