ITU CIVIL ENGINEERING FACULTY HYDRAULICS DIVISION PROBABILITY and STATISTICS Fall 2016 HOMEWORK - I I. The road linking A to C passes through B. The time of travel T1 from A to B has the following probability mass function: P(T1=1)=0,2 P(T1=2)=0,4 P(T1=3)=0,3 P(T1=4)=0,1 The time of travel T2 from B to C has the following probabilities: P(T2=4)=P(T2=6)=0,25 P(T2=5)=0,5 a) Determine the probability mass function of T, time of travel from A to C. b) Compute the mean and standard deviation of T. c) What is the probability that T>8 ? II. The traffic density X for the design of a highway is assumed to have a triangular probability density function between 0 and 400 vehicles/hr with the peak at 300 vehicles/hr . Compute several estimates of the design density as: a) b) c) d) e) The mode of X. The mean of X. The median of X. The value of X with the probability of exceedance 10% For each case compute the probabilities that the design capacity will be exceeded. III. When three lanes are used in one direction the capacity of a highway is 100 vehicles/min . In the rush time two more lanes are added from the opposite direction, increasing the capacity to 140 vehicles/min . In normal traffic the density has a triangular probability density function between 0 and 150 vehicles/min with the peak at 75 vehicles/min . Rush time traffic density is also triangular between 0 and 200 vehicles/min with the peak at 100 vehicles/min . Along a day the probability of normal traffic is twice the probability of rush hour traffic. Compute the probability that the capacity of the road is inadequate. IV. The joint PMF of precipitation, X (cm) and runoff, Y (m3/s) (discretized here for simplicity) due to storms at a given location as as follows: Y=10 Y=20 Y=30 a) X=1 0.05 0.10 0.0 X=2 0.15 0.25 0.10 X=3 0.0 0.25 0.10 What is the probability that the next storm will bring a precipitation of 2 or more cm and a runoff more that 20 m3/s. b) After a storm, the rain gauge indicates a precipitation of 2 cm. What is the probability that the runoff in this storm is 20 m3/s or more? c) Are X and Y statistically independent? Substantiate your answer. d) Determine and plot the marginal PMF of runoff e) Determine and plot the PMF of runoff for a storm whose precipitation is 2 cm. f) Determine the correlation coefficient between precipitation and runoff