210k views
1 vote
Find the mathematical expectation of a random variable with

(a) uniform distribution over the interval [a,b],
(b) triangle distribution,
(c) exponential distribution.

1 Answer

2 votes

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.

User Jjei
by
8.0k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories