Given a continuous uniform distribution, show that (a) μ=(A+B)/(2), and (b) σ^2-((B-λ)^2)/(12)

Name: Problem 20, Chapter 6: Probability and Statistics for Engineers and Scientists
Uploaded: 2022-01-22T22:13:03-08:00
Duration: 5 min 29 s
Channel: Raymond Matshanda
Description: Given a continuous uniform distribution, show that (a) μ=(A+B)/(2), and (b) σ^2-((B-λ)^2)/(12)

step 1 All right, this question says, given a continuous distribution, show that a mu is equal to a plu

Given a continuous uniform distribution, show that (a)...

Key Concepts

Continuous Uniform Distribution

The continuous uniform distribution is a probability distribution where all outcomes in a given interval are equally likely. This means that the probability density function (PDF) is constant over the interval and zero outside. It serves as a basic model in probability theory for situations where there is complete uncertainty about where within the interval the outcome might fall.

Expected Value (Mean) Calculation

The expected value (or mean) of a continuous random variable is computed using the integral of the variable multiplied by its probability density function over its entire range. For a continuous uniform distribution defined on an interval, the symmetry of the interval directly leads to the result that the mean is the midpoint of the interval. This property makes it straightforward to derive that the mean is given by (A + B)/2 for an interval [A, B].

Variance Calculation

Variance measures the spread of a distribution around its mean and is calculated as the expected value of the squared deviation from the mean. For the continuous uniform distribution, integrating the squared difference from the mean over the uniform interval and normalizing by the interval length yields the well-known formula for variance, (B - A)²/12. This result reflects the uniform spread of the distribution over the interval and is derived using standard techniques of calculus.

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