Question

If x and y are independent random variables then e(xy)is equal to

Answer 1

e is the constant

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Answer 2

If X and Y are independent random variables then E(XY) = E(X) E(Y)

Given :

X and Y are independent random variables

To find :

The value of E(XY)

Solution :

Step 1 of 2 :

Write down the given data

Here it is given that X and Y are independent random variables

Let f(x, y) be the density function.

Since X and Y are independent random variables

$\displaystyle \sf{ \therefore \: \:f(x, y) = f_x(x) f_y(y) }$

Step 2 of 2 :

Find the value of E(XY)

$\sf E(XY)$

$\displaystyle = \int\limits_{- \infty}^{\infty} \int\limits_{- \infty}^{\infty} \sf xy \: f(x,y)\, dx \: dy$

$\displaystyle = \int\limits_{- \infty}^{\infty} \int\limits_{- \infty}^{\infty} \sf xy \: f_x(x)f_y(y)\, dx \: dy$

$\displaystyle \sf{ = \int\limits_{- \infty}^{\infty} \sf x \: f_x(x)\, dx \: \times \int\limits_{- \infty}^{\infty} \sf y \: f_y(y)\, dx }$

$\displaystyle \sf{ = E(X) E(Y) }$

Hence when X and Y are independent random variables then E(XY) = E(X) E(Y)

Correct question : If X and Y are independent random variables then E(XY) equals to

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