# Answer to Question #4411 in Statistics and Probability for miya

Question #4411

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?

Expert's answer

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

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

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