USE EXCEL TO SLOVE THIS AND SHOW THE FORMULAS FOR EASY DUPLICATION. The joint pr

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USE EXCEL TO SLOVE THIS AND SHOW THE FORMULAS FOR EASY DUPLICATION.

The joint probability distribution of the number X of cars and the number Y of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table p(x, y) 0 0.010 0.015 0.025 1 0.020 0.030 0.050 2 0.050 0.075 0.125 3 0.060 0.090 0.150 4 0.040 0.060 0.100 5 0.020 0.030 0.050 (a) What is the probability that there is exactly one car and exactly one bus during a cycle? (b) What is the probability that there is at most one car and at most one bus during a cycle? 0.17x (c) What is the probability that there is exactly one car during a cycle? Exactly one bus? P(exactly one car)0.1 P(exactly one bus)-p (d) Suppose the left-turn lane is to have a capacity of five cars and one bus is equivalent to three cars. What is the probability of an overflow during a cycle? 0.62 X e) Are X and Y independent rv's? Explain @ Yes, because ?(x, y)-PX(x)-Pr(y) O Yes, because ?(x, y) # PX(x)-pr(y) O No, because p(x, y)Px(x) pyty) O No, because p(x, y)px(x) py(y)

Explanation / Answer

We need to give answer for b and d

Let's draw table first

b.) We need to find probabaility for at most 1 car and at most 1 bus during a cycle

So, reauired table looks like

Summing up all the probabailities which covers all possible combinations.

The combinations which are included are:

-> 0 Car and 0 Bus

-> 0 Car and 1 Bus

-> 1 Car and 0 Bus

-> 1 Car and 1 Bus

So, P = 0.015 + 0.02 + 0.03 + 0.015 = 0.075

d.) Left turn has capacity of 5 Cars only

We need to find probabaility in case of Over flow

Also, 1 bus = 3 Cars

So, all possible combinations for overflow are:

-> 1 Bus and 3 Cars

-> 1 Bus and 4 Cars

-> 1 Bus and 5 Cars

-> 2 Bus will lead to over flow irrespective number of cars

So matrix looks like:

Summing up all above probabailities we get,

P = 0.68

x,y 0 1 2 0 0.01 0.015 0.025 1 0.02 0.03 0.05 2 0.05 0.075 0.125 3 0.06 0.09 0.15 4 0.04 0.06 0.1 5 0.02 0.03 0.05

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