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.
What is the probability that w is less than 50? Less than 25? Greater than 75?What is the probability that w is between 25 and 75?
No idea help
To find the probability that w is less than 50, we need to calculate the area under the density curve from 0 to 50.
The density curve is a triangle with a base of length 2a (in this case, 2*50 = 100) and a height of b (0.02). The area of a triangle is given by the formula: 1/2 * base * height. So the area of this triangle is 1/2 * 100 * 0.02 = 1.
Since the total area under the density curve represents the probability of all possible values, the probability of w being less than 50 is equal to the area of the triangle, which is 1.
To find the probability that w is less than 25, we need to calculate the area under the density curve from 0 to 25. Since the density curve is a triangle, we can use a similar approach as above to find the area of a smaller triangle.
Using the formula for the area of a triangle, the area of the triangle with a base of 50 (half of the base of the larger triangle) and a height of 0.02 is: 1/2 * 50 * 0.02 = 0.5.
Therefore, the probability that w is less than 25 is 0.5.
To find the probability that w is greater than 75, we need to calculate the area under the density curve from 75 to 100 (the maximum possible value of w). Since the density curve is a triangle, we can subtract the area of the triangle from 0 to 75 from the total area to get the desired probability.
The area of the triangle from 0 to 75 is 1, as we calculated earlier. So the area from 75 to 100 is 1 - 1 = 0.
Therefore, the probability that w is greater than 75 is 0.
To find the probability that w is between 25 and 75, we need to calculate the area under the density curve between those two values. Since the density curve is a triangle, we can subtract the area of the triangle from 0 to 25 from the area of the triangle from 0 to 75.
The area of the triangle from 0 to 25 is 0.5, as we calculated earlier. The area of the triangle from 0 to 75 is 1. So the area between 25 and 75 is 1 - 0.5 = 0.5.
Therefore, the probability that w is between 25 and 75 is 0.5.