Question

Recently a study was performed on a Queenslad highway regarding the quantity and speed of vehicles that use the road. It is your job to determine the probability of certain events with the use of probability distributions. For each question, clearly state which probability distribution the random variable follows. You must solve each problem by hand and verify using a MATLAB function.

During the testing time, it was found that cars pass by at an average rate of 300 per hour, with the time between cars being independent. What is the probability of seeing less than 5 cars in a minute?

What is the probability of there being more than 1 minute between cars?

The speed of cars was found to follow a normal distribution with a mean of 95 km/h and a variance of 5 km2/h2 . Given the speed limit is 100km/h, what is the probability that a random car is speeding?

At any given point, what is the probability that at least 2 out of the next 5 cars will be speeding?

Answer 1

The number of cars seen in a minute has Poisson distribution.

The Poisson PMF is Here .

The probability of seeing less than 5 cars in a minute is

Use MATLAB function

poisscdf(4,5) = 0.4404933

The time between cars is exponetially distributed with . The PDF is

The probability of there being more than 1 minute between cars is

Use MATLAB function

1 - expcdf(1,1/5) = 0.00674

The speed of cars was found to follow a normal distribution .

The probability that a random car is speeding is

The number orf cars out of 5 speeding has Binomial distribution.The PMF is

. Here

The probability that at least 2 out of the next 5 cars will be speeding is