Answer to Question #4411 in Statistics and Probability for miya

Question

suppose that the random variable X has a uniform distribution on interval [0,1].random variable Y has a uniform distribution on the interval [4,10].X and Y are independent.suppose a rectangle is to be constructed for which the length of two adjacent sides are X and Y.so what is the expected value of area of this rectangle?

Answer 1

Area of a rectangle is a random variable S=X*Y. Let's use the formula:
E(XY)
= E(X)*E(Y) + COV(X,Y).
As X and Y are independent, then COV(X,Y) = 0
and
E(XY) = E(X)*E(Y)
Let's use the property: if a random variable Z has a
uniform distribution on [a, b], then E(Z) = (a + b) / 2.
So, E(X) = 1/2 and
E(Y) = 7.
Consequently, E(S) = E(XY) = 7/2 = 3.5