Final answer:
The mathematical expectation, or expected value, of a random variable varies based on the distribution: for a uniform distribution over [a,b] it is (a + b)/2, for a triangle distribution with limits a, b, and mode c it is (a + b + c)/3, and for an exponential distribution with rate \( \lambda \) it is 1/\( \lambda \).
Step-by-step explanation:
The mathematical expectation of a random variable, also known as the expected value, is a fundamental concept in probability and statistics. It represents the long-run average value of repetitions of the experiment it represents. Here's how you find it for different distributions:
For a uniform distribution over the interval [a,b], the expected value E(X) is given by:
E(X) = \( \frac{a + b}{2} \)
For a triangular distribution, if the lower limit is 'a', the upper limit is 'b', and the mode (or peak) is 'c', the expected value E(X) is given by:
E(X) = \( \frac{a + b + c}{3} \)
For an exponential distribution with rate parameter \( \lambda \), the expected value E(X) is given by:
E(X) = \( \frac{1}{\lambda} \)
These expected values are derived from the respective probability density functions for each distribution, and they provide a measure of the 'central location' of the distribution.