Determine the solutions to Problem 3.59 by MCS. The joint PMF of precipitation,

Question

Determine the solutions to Problem 3.59 by MCS. The joint PMF of precipitation, X(in.) and runoff, y(cfs) (discretized here for simplicity) due to storms at a given location is as follows: (a) What is the probability that the next storm will bring a precipitation of 2 or more in. and a runoff of more than 20 cfs? (b) After a storm, the rain gauge indicates a precipitation of 2 in. What is the probability that the runoff in this storm is 20 cfs 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 in. (f) Determine the correlation coefficient between precipitation and runoff.

Explanation / Answer

(a)

Here we are left with only two cases: X=2,Y=30 and X=3,Y=30

Thus, P(X>=2 , Y>30) = 0.10 + 0.10 = 0.20

(b)

Here we are left with only two cases: X=2,Y=20 and X=2,Y=30

Thus, P(X=2 , Y>=20) = 0.25 + 0.10 = 0.35

(c)

No the two random variables X and Y are not statistically independant since happening of one event depends on the occurrence of other. They are correlated.

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