Question

Given an arrival process with λ=0.7, what is the probability that an arrival occurs in the first t=5 time units? P(t≤5∣λ=0.7)= (Round to four decimal places as needed.)

Answer 1

To find the probability that an arrival occurs within the first 5 time units given an arrival rate (λ) of 0.7, we use the exponential distribution's CDF. The calculated probability is 0.9698 after rounding to four decimal places.

Step-by-step explanation:

The question asks for the probability that an arrival occurs within the first t=5 time units given an arrival rate (λ) of 0.7. This can be assessed using the exponential distribution, which describes the time between events in a Poisson process. The cumulative distribution function (CDF) of the exponential distribution is given by P(T < t) = 1 - e-λt. To find the probability that an arrival occurs within the first 5 units of time, we need to calculate P(t ≤ 5 | λ=0.7).

The calculation is as follows:

P(t ≤ 5 | λ=0.7) = 1 - e-0.7 × 5

= 1 - e-3.5

= 1 - 0.03019738

= 0.96980262

After rounding to four decimal places, we get P(t ≤ 5 | λ=0.7) = 0.9698.