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 = 10, b = 0.1.
Less than 5?
Greater than 15?
What is the probability that w is between 5 and 15? (Hint: It might be easier first to find the probability that w is not between 5 and 15.)
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To find the probabilities for the given values of a and b in the triangular distribution, we can use the properties of the density curve.
1. Probability that w is less than 5:
To find this probability, we need to calculate the area under the curve to the left of 5. Since the triangular distribution forms a triangle, we can find the area of the triangle formed by the line connecting (0, 0) to (a, 1) and (a, 1) to (2a, 0). The area of a triangle is given by the formula: (base * height) / 2.
The base of the triangle is 2a, and the height is 1. Plugging in the values a = 10, we get: (2 * 10 * 1) / 2 = 20 / 2 = 10.
Therefore, the probability that w is less than 5 is 10.
2. Probability that w is greater than 15:
To find this probability, we need to calculate the area under the curve to the right of 15. Since the triangular distribution forms a triangle, we can find the remaining area under the curve, which is the area of the triangle formed by the line connecting (0, 0) to (a, 1), (a, 1) to (15, 0), and (15, 0) to (2a, 0).
The base of the remaining triangle is 15 - a, and the height is 1. Plugging in the values a = 10, we get: (15 - 10) * 1 = 5.
Therefore, the probability that w is greater than 15 is 5.
3. Probability that w is between 5 and 15:
To find this probability, we subtract the probability that w is less than 5 from the probability that w is greater than 15. We already calculated these probabilities, which were 10 and 5 respectively.
Therefore, the probability that w is between 5 and 15 is 10 - 5 = 5.
Note: The hint suggests finding the probability that w is not between 5 and 15 first. This can be done by subtracting the probability that w is between 5 and 15 from 1. In this case, 1 - 5 = -4, which does not make sense as a probability value. Therefore, finding the probability that w is between 5 and 15 directly is the correct approach.
To answer these questions, we can refer to the triangular distribution shown for w. Given that a = 10 and b = 0.1, we can proceed with the calculations.
1. Less than 5:
To find the probability that w is less than 5, we need to calculate the area under the density curve up to the value of 5. Since the density curve is a triangle, we can compute the area as ½(base × height). In this case, the base is 10 and the height is 0.2 (since b = 0.1). Thus, the area under the curve up to 5 is ½(10 × 0.2) = 1.
Therefore, the probability that w is less than 5 is 1.
2. Greater than 15:
To find the probability that w is greater than 15, we need to calculate the area under the density curve beyond the value of 15. Since the density curve is a triangle, we can compute the area as ½(base × height). In this case, the base is 10, but to determine the height, we need to calculate the value of b at 15.
Since x and y can range from 0 to a, the maximum value of x and y is 10. Therefore, w can be maximum when x = 10 and y = a, resulting in w = 20. To find the height at 15, we can calculate the value of b at x = 10 by using the formula b = (2/a)(1 - x/a). Plugging in the values, we have b = (2/10)(1 - 10/10) = 0.2.
Thus, the height of the triangle at x = 10 (which corresponds to w = 15) is 0.2. Therefore, the area under the curve beyond 15 is ½(10 × 0.2) = 1.
Therefore, the probability that w is greater than 15 is 1.
3. The probability that w is between 5 and 15:
To find this probability, we can subtract the probabilities obtained above from 1 since it is easier to compute the probability that w is not between 5 and 15.
The probability that w is not between 5 and 15 is the sum of the probabilities that w is less than 5 and that w is greater than 15. Therefore, the probability that w is between 5 and 15 is 1 - (1 + 1) = 1 - 2 = -1.
However, it is not possible for a probability to be negative. Therefore, the probability that w is between 5 and 15 is 0 (or impossible) based on the given parameters.
In summary:
Less than 5: Probability = 1
Greater than 15: Probability = 1
Between 5 and 15: Probability = 0