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) =

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

P(w < 8) = Well, I once waited for 8 minutes for a train and it felt like an eternity. So, the probability of w being less than 8 is as close to zero as the chance of finding a pot of gold at the end of a rainbow.

Less than 4?
P(w < 4) = Let's see, if x and y are both less than 4, then the maximum value of w can be 8. But since we're looking for w to be less than 4, it means both x and y have to be less than 4. So, the probability is like finding a Unicorn in a petting zoo - extremely rare!

Greater than 12?
P(w > 12) = Well, if w is greater than 12, it means you've waited for more than 12 minutes for the train on both days. That's like waiting for a sequel to a bad movie - highly unlikely!

(b) What is the probability that w is between 4 and 12?
P(4 < w < 12) = To find the probability that w is between 4 and 12, we can first find the probability that w is not between 4 and 12, and then subtract it from 1.

The probability that w is not between 4 and 12 is like trying to find a parking spot in a crowded mall during the holiday season - it's a nightmare! So, the probability of w being between 4 and 12 is 1 minus the probability of finding a parking spot in that mall – extremely low!

To find the probabilities, we need to calculate the areas under the density curve of the triangular distribution.

(a)
To find the probability that w is less than 8, we need to calculate the area under the curve up to 8.

P(w < 8) = (8 - 0) * (8/2) * (0.125) = 4 * 4 * 0.125 = 2

Therefore, P(w < 8) = 2.

To find the probability that w is less than 4, we need to calculate the area under the curve up to 4.

P(w < 4) = (4 - 0) * (4/2) * (0.125) = 2 * 2 * 0.125 = 0.5

Therefore, P(w < 4) = 0.5.

To find the probability that w is greater than 12, we need to subtract the area under the curve up to 12 from the total area under the curve (triangle).

Total area under the curve = (8 - 0) * (8/2) * (0.125) = 2
Area under the curve up to 12 = (12 - 0) * (12/2) * (0.125) = 3 * 6 * 0.125 = 2.25

P(w > 12) = Total area - Area up to 12 = 2 - 2.25 = -0.25

Since probability cannot be negative, P(w > 12) = 0.

(b) To find the probability that w is between 4 and 12, we can subtract the probability that w is not between 4 and 12 from 1.

P(4 < w < 12) = 1 - (P(w < 4) + P(w > 12))
= 1 - (0.5 + 0)
= 1 - 0.5
= 0.5

Therefore, P(4 < w < 12) = 0.5.

To answer these questions, we need to use the probability density function (PDF) associated with the triangular distribution.

(a) What is the probability that w is less than 8?
To find P(w < 8), we need to calculate the area under the density curve from 0 to 8. The triangular distribution has a maximum value at the midpoint of the interval (a), which in this case is 8. So, the area of the triangle can be calculated as (base * height) / 2. The base is 8 (the length of the interval), and the height is 1/8 (since the maximum value is 8 and the area is 1). Therefore, the area is (8 * 1/8) / 2 = 1/2. Thus, P(w < 8) = 1/2.

Less than 4?
To find P(w < 4), we again need to calculate the area under the density curve, this time from 0 to 4. Since 4 is less than the midpoint (8), the area is a triangle with a base of 4 and a height of 1/8. Therefore, P(w < 4) = (4 * 1/8) / 2 = 1/4.

Greater than 12?
To find P(w > 12), we can use the complement rule. P(w > 12) is equal to 1 minus the probability that w is less than or equal to 12. The area under the density curve from 0 to 12 is a triangle with a base of 12 and a height of 1/8. Therefore, P(w > 12) = 1 - ((12 * 1/8) / 2) = 1 - 3/4 = 1/4.

(b) What is the probability that w is between 4 and 12?
To find P(4 < w < 12), we need to calculate the area under the density curve between these two values. We can first find the area up to 12 (calculated above) and subtract the area up to 4. This means P(w < 4) + P(w > 12) = 1/4 + 1/4 = 1/2. Therefore, P(4 < w < 12) = 1 - 1/2 = 1/2.