Question

Let X have a uniform distribution on the interval [a, b]. Obtain an expression for the (100p) th percentile. Compute E(X), V(X), and sigma_2. For n a positive integer, compute E(X^n)

Answer 1

To obtain the (100p)th percentile of a uniform distribution, the formula is x = a + p(b - a). The mean, variance, and standard deviation of X can be calculated using formulas. For the expected value of X^n, a formula is provided.

Step-by-step explanation:

To obtain the (100p)th percentile of a uniform distribution on the interval [a, b], you can use the formula x = a + p(b - a). For example, if you want to find the 50th percentile (median), you can substitute p = 0.5 into the formula.

The expected value or mean of X is given by E(X) = (a + b) / 2. The variance of X is given by V(X) = (b - a)^2 / 12, and the standard deviation is sigma = sqrt(V(X)).

For a positive integer n, the formula for the expected value of X^n is E(X^n) = (1/n+1) * (b^(n+1) - a^(n+1)) / (b - a).