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 􀀀 e6􀀀x=10, and if you
are indeed stopped you will pay the �ne of 100(x 􀀀 60) dollars. Find the probability you are
stopped and pay more than 1000 dollars.

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 calculate two probabilities:

1. Probability of being stopped: We are given that the speed is exponentially distributed with a mean of 50 miles per hour. The probability of being stopped is given by the cumulative distribution function (CDF) of the exponential distribution. The CDF of an exponential distribution with mean λ is given by 1 - e^(-λx). In this case, λ = 1/50 since the mean is 50 miles per hour. So, the probability of being stopped is 1 - e^(-x/50).

2. Probability of paying more than 1000 dollars: We are also given that if you are stopped, the fine you have to pay is 100(x - 60) dollars. To pay more than 1000 dollars, the fine amount (100(x - 60)) needs to be greater than 1000. So we have the inequality: 100(x - 60) > 1000. Simplifying this inequality gives us x > 70.

Now, to find the probability you are stopped and pay more than 1000 dollars, we need to calculate the intersection of these two probabilities:

P(x > 70 and stopping) = P(x > 70) * P(stopping)

P(x > 70) = 1 - e^(-70/50) = 1 - e^(-7/5) ≈ 0.736

P(stopping) = 1 - e^(-x/50)

Therefore,

P(x > 70 and stopping) = (1 - e^(-7/5)) * (1 - e^(-x/50))

Please note that this calculation assumes the speed and the stopping probability are independent random variables.