Let x and y be the amounts of time (in minutes) that a particular commuter must

Question

Let x and y be the amounts of time (in minutes) that a particular commuter must wait for a train on two independently selected days. Define a new random variable w by w = x + y, the sum of the two waiting times. The set of possible values for w is the interval from 0 to 2a (because both x and y can range from 0 to a). It can be shown that the density curve of w is as pictured (this curve is called a triangular distribution, for obvious reasons!)
Answer the following questions assuming a = 20, b = 0.05.

(a) What is the probability that w is less than 20?
P(w < 20) =

Less than 10?
P(w < 10) =

Greater than 30?
P(w > 30) =

(b) What is the probability that w is between 10 and 30? (Hint: It might be easier first to find the probability that w is not between 10 and 30.)
P(10 < w < 30) =

Explanation / Answer

a) P(W<20)= 0.5 (as 20 lies in midway of both the traingle and it is symmetric)

b)P(w<10)=0.5/4=0.125(as height at w=10 is half at hight at 20 and base is also half of 20 hence area is 1/4 of w<20)

c)similarly P(w>30)=0.125

d)P(10<W<30) =P(W<30)-P(W<10)=1-0.125-0.125=0.75

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