10. The times between train arrivals at a certain train station is exponentially
ID: 3047905 • Letter: 1
Question
10. The times between train arrivals at a certain train station is exponentially distributed with a mean of 10 minutes. I arrived at the station while Dayer was already waiting for the train. If Dayer had already spent 8 minutes before I arrived, determine the following a. The average length of time I wil wait until the next train arrives b. The probability that I will wait more than 5 minutes until the next train arrives c. The probability that I will wait between 3 and 6 minutes until the next train airives d. Assume that I came to conduct a study on the inter-arrival times of trains at that station. Let Y be a random variable that denotes the time from the instant thatI arrived until the arrival of the 4th train. (i) What type of random variable is Y? ii) What is the PDF of Y? ii) What is the expected value of Y? (iv) What is the variance of Y?Explanation / Answer
Question 10
The first thing we shall clear that esxponential distribution has memoryless propoerty so in this question, as Dayer had waited for more than 8 minutes will not effect any of the results here or on the distribution of my waiting time.
(a) Average length of time until the next train arrives E(T) = 10 minutes
(b) Here
f(t) = (1/10)e-t/10
F(t) = 1 - e-t/10
Pr( t > 5 minutes) = 1 - Pr(t < 5 minutes) = 1 - (1 - e-5/10) = 0.6065
(d) Pr(3 mins < t < 6 mins) = Pr(t < 6 mins) - Pr(t < 3 mins) = (1 - e-6/10) - (1 - e-3/10) = 0.74082 - 0.5488 = 0.1920
(d) Here as Y = the time from the instant I arrived until the arrival of the 4th train.
(i) Here random variable Y is exponetial variable (interval variable) and in units of time
(ii) Here E(Y) = 4 * E(X) = 40 mins
f(Y) = (1/40) e-y/40
(iii) Expeced value of Y = 40 mins
(iv) Variance of Y = Var(Y) = 402 = 1600
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.