You are driving a car and the speed is an exponentialy distributed random variable
with mean 50 (in miles per hour). The speed limit is 60 miles per hour, and if your speed is
x > 60 miles per hour, the probability you are stopped by the police is 1 - e^(6-(10/x)), and if you
are indeed stopped you will pay the fine of 100(x - 60) dollars. Find the probability you are
stopped and pay more than 1000 dollars.
To find the probability that you are stopped and pay more than 1000 dollars, we need to find the probability that your speed is greater than 60 miles per hour and the fine you would pay is more than 1000 dollars.
Let's break down the problem into two steps:
Step 1: Find the probability that your speed is greater than 60 miles per hour.
Given that the speed is exponentially distributed with a mean of 50 miles per hour, we can use the exponential probability density function (PDF) to find this probability. The PDF for an exponential distribution with mean parameter λ is given by:
f(x) = λ * e^(-λx)
In this case, the mean speed is 50 miles per hour, so the rate parameter λ is equal to 1/50. We want to find the probability of having a speed greater than 60, so we integrate the PDF from 60 to infinity:
P(speed > 60) = ∫[60, ∞] λ * e^(-λx) dx
Integrating this expression will give us the probability that the speed is greater than 60 miles per hour.
Step 2: Find the probability that the fine you would pay is more than 1000 dollars.
Using the formula given, we know that the fine you would pay is 100(x - 60) dollars, where x is the speed in miles per hour. We want to find the probability that this fine is more than 1000 dollars. Therefore, we need to find the values of x for which the fine exceeds 1000:
100(x - 60) > 1000
x - 60 > 10
x > 70
So, we want to find the probability that the speed is greater than 70 miles per hour.
Finally, to find the probability you are stopped and pay more than 1000 dollars, we multiply the probabilities obtained in Step 1 and Step 2 since both events need to occur simultaneously:
P(stopped and pay > 1000) = P(speed > 60) * P(speed > 70)
By calculating these probabilities, you can find the answer to your question. However, please note that this answer is based on the information provided, and the actual probabilities may depend on additional factors or assumptions.