Question

Suppose the proportion X of surface area in a randomly selected quadrat that is covered by a certain plant has a standard beta distribution with α = 5 and β = 3.
(a) Compute E(X) and V(X).
(b) Compute P(X ? 0.2).
(c) Compute P(0.2 ? X ? 0.6).
(d) What is the expected proportion of the sampling region not covered by the plant?

Answer 1

Answer:

(a) E(X)  = 0.625

    V(X) = 0.0260

(b) P(X≤0.2) = 0.00468

(c) P(0.2≤X≤0.6) =0.425

(d) The expected proportion = 0.375

Step-by-step explanation:

Given data;

α = 5

β = 3

(a) E(X) and V(X).

the mean E(X) of a beta distribution is given by the formula;

E(X) = α/(α+β)

Substituting, we have;

E(X) = 5/(5+3)

        = 5/8

        = 0.625

The variance V(X) of a beta distribution is given by the formula;

V(X) = αβ/(α+β)²(α+β+1)

Substituting, we have;

V(X) = 5*3/(5+3)²(5+3+1)

       = 15/(8² * 9)

       = 15/576

      = 0.0260

See the attached file for solution to (b), (c) and (d