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 = 8, b = 0.125.

(a) What is the probability that w is less than 8?
P(w < 8) =  

Less than 4?
P(w < 4) =  

Greater than 12?
P(w > 12) =  

(b) What is the probability that w is between 4 and 12? (Hint: It might be easier first to find the probability that w is not between 4 and 12.)
P(4 < w < 12) =

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 (a) What is the probability that w is less than 8? P(w

Explanation / Answer

a)

P(w <8) = 1/2 * 8 * 0.125 = 0.5

as we can height of pdf is decreasing at a same rate as length

P(w<4) = 1/2 * 4 * 0.125/2 = 0.125

P(w>12) = 1/2 * (16-12) * 0.125/2 = 0.125

b)

P(4<w<12) = 1- P(w<4) - P(w>12)

= 1 - 0.125 - 0.125

= 0.75

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