ECE2191 Probability Models in Engineering
Tutorial 5: Continuous Random Variables
Second semester 2021
1. Trains headed for destination A arrive at the train station at 15-minute intervals starting at 7 A.M., whereas trains headed for destination B arrive at 15-minute intervals starting at 7:05 A.M.
(a) If a certain passenger arrives at the station at a time uniformly distributed between 7 and 8 A.M. and then gets on the first train that arrives, what proportion of time does he or she go to destination A?
(b) What if the passenger arrives at a time uniformly distributed between 7:10 and 8:10 A.M.?
2. A point is chosen uniformly at random on a line segment of length L.
(a) What is the PDF of the position of the point. Take the left endpoint of the line as the origin.
(b) Find the probability that the ratio of the shorter to the longer segment is less than 1/4.
3. The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by
(a) What is P(X > 15)?
(b) What is the CDF of X?
(c) What is the probability that, of 6 such types of devices, at least 3 will function for at least 15 hours? What assumptions are you making?
4. You arrive at a bus stop at 10 o’clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30.
(a) What is the probability that you will have to wait longer than 10 minutes?
(b) If, at 10:15, the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?
5. Calamity Jane goes to the bank to make a withdrawal, and is equally likely to find 0 or 1 customers ahead of her. The service time of the customer ahead, if present, is exponentially distributed with parameter λ . What is the CDF of Jane’s waiting time?
6. Let X be a continuous random variable with PDF
Find E[Xn], where n ∈ N.
7. Let X be a continuous random variable with PDF
Find the mean and variance of X .
8. Let X be a continuous uniform random variable on [0, 10].
(a) Find the mean and variance of X
(b) Calculate the probabilities P(X - 5 ≥ 4) and P(jX - 5j ≥ 4).
9. The time, T , that is spent on a key step in a manufacturing process follows an exponential distribution with mean 65 hours. The cost of completing this key step is X = 1000 + 500T dollars. Determine the mean and the standard deviation of X .
10. The alpha decay of Actinium-220 atoms to Francium-216 is a random process with a half life, γ1/2, of 26 ms. The time taken for any individual atom to decay can be modelled using an exponentially distributed random variable, T. Give an expression for the cumulative probability distribution function for this variable and find the parameter, λ, that appears in this function. (Note: half life is the time taken for the number of Actinium-220 atoms in the sample to halve.
11. [Optional] Suppose that an average of 30 customers per hour arrive at a store and the time between arrivals is exponentially distributed.
(a) On average, how many minutes elapse between two successive arrivals?
(b) When the store first opens, how long on average does it take for three customers to arrive?
(c) After a customer arrives, find the probability that it takes less than one minute for the next customer to arrive.
(d) After a customer arrives, find the probability that it takes more than five minutes for the next customer to arrive.
(e) Find the time (minutes) such that the probability of the next customer arriving within that time is 0.7?
(f) Is an exponential distribution reasonable for this situation?