Question 1195988: Suppose the number of cars that arrive at a car wash during one hour is described by a Poisson probability distribution with a mean of 6 cars per hour. Now we are interested in the time, denoted by X, between the arrivals.
1. What is probability distribution of X?
2. What is probability that arrival time is greater than 10 minutes?
Answer by proyaop(69)   (Show Source):
You can put this solution on YOUR website! **1. Probability Distribution of X**
* **If the number of arrivals per hour follows a Poisson distribution, then the time between arrivals follows an Exponential distribution.**
**2. Probability that Arrival Time is Greater Than 10 Minutes**
* **Convert minutes to hours:** 10 minutes = 10/60 hours = 1/6 hour
* **Exponential Distribution:**
* The probability density function (PDF) of an exponential distribution is:
* f(x) = λ * e^(-λx)
* where:
* λ is the rate parameter (average number of arrivals per unit time) = 6 cars/hour
* x is the time between arrivals (in hours)
* **Cumulative Distribution Function (CDF):**
* The probability of the time between arrivals being less than or equal to 't' is given by:
* F(t) = 1 - e^(-λt)
* **Probability of Arrival Time Greater Than 10 Minutes:**
* P(X > 1/6) = 1 - P(X ≤ 1/6)
* P(X > 1/6) = 1 - F(1/6)
* P(X > 1/6) = 1 - (1 - e^(-6 * (1/6)))
* P(X > 1/6) = 1 - (1 - e^(-1))
* P(X > 1/6) = e^(-1)
* P(X > 1/6) ≈ 0.3679
**Therefore, the probability that the arrival time is greater than 10 minutes is approximately 0.3679 or 36.79%.**
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