Suppose X is uniformly distributed on the interval a to b. Show that E[X]=(a+b)/(2) and σ^2=((

step 1 So in this question, x is uniformly distributed on the interval 1 to 2. And that means that the

Suppose X is uniformly distributed on the interval a to b....

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Key Concepts

Uniform Distribution

A continuous probability distribution where all outcomes in a given interval have equal likelihood. The density function is constant across the interval, and this simplicity allows for straightforward computation of probabilities, expectations, and variances using integrals.

Expected Value

The expected value, or mean, of a random variable represents its average or central value. For continuous distributions, it is computed by integrating the product of the variable and its probability density function over the entire range, providing a measure of the center of mass of the distribution.

Variance

Variance measures the spread or dispersion of a random variable around its mean. It is defined as the expected value of the squared difference between the variable and its mean, calculated as E[X²] minus the square of E[X]. This metric quantifies how much the values of the variable differ from the average value.

Integration in Probability

Integration is a fundamental tool in probability theory for computing the moments of continuous distributions. By integrating the probability density function multiplied by a function of the random variable, one can derive key statistical measures such as the mean and variance.

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