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 = 50, b = 0.02.

(a) What is the probability that w is less than 50?
P(w < 50) =  

Less than 25?
P(w < 25) =  

Greater than 75?
P(w > 75) =  

(b) What is the probability that w is between 25 and 75? (Hint: It might be easier first to find the probability that w is not between 25 and 75.)
P(25 < w < 75) =

Explanation / Answer

Solution:

a)

P(w < 50) = 1/2 * 50 * 0.02 = 0.5

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

P(w<25) = 1/2 * 25 * 0.02/2 = 0.125

P(w>75) = 1/2 * (100-75) * 0.125/2 = 0.781

b)

P(25<w<75) = 1- P(w<25) - P(w>75)

= 1 - 0.125 - 0.781

= 0.094

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.