Final answer:
A triangular probability distribution is defined by its minimum 'a', maximum 'b', and mode 'm'. For the given values of a = 50, b = 375, and m = 250, the student will sketch a triangular shape on a graph with the specified points as the corners and ensure that the total area under the triangle equals 1.
Step-by-step explanation:
The student has been given parameters for a triangular probability density function (pdf) and is asked to sketch the probability distribution. In a triangular distribution, 'a' is the minimum, 'b' is the maximum, and 'm' is the mode of the distribution. Given a = 50, b = 375, and m = 250, the distribution will be a triangle with the left corner at the point (50, 0), the peak at the point (250, height), and the right corner at the point (375, 0). The height can be calculated using the formula for the area of a triangle (Area = 1), leading to the height being 2/((b-a) * (m-a)).
The probability density function is then defined as follows:
- For 50 ≤ x < 250, the slope will be upward from (50, 0) to (250, height).
- For 250 ≤ x ≤ 375, the slope will be downward from (250, height) to (375, 0).
The total area under the curve must equal 1, as this is a probability density function. This area represents the cumulative probability from 'a' to 'b'. Since this is a basic geometry shape, the area can be verified simply by calculating the area of the triangle formed by the points a, m, and b.