Question

Suppose \(X\) and \(Y\) are independent continuous random variables. Show that $$ E[X \mid Y=y]=E[X] \text { for all } y $$

Short answer

Using the definition of conditional expectation and the property of independence between random variables X and Y, we can find the conditional probability density function of X given Y = y. We then substitute this back into the definition of conditional expectation and recognize that the integral in the expression is the expected value of X. Thus, for independent continuous random variables X and Y, the expected value of X given Y equals y is the same as the expected value of X: \(E[X \mid Y=y] = E[X]\). This holds for all y.

Step-by-step solution

Write down the definition of conditional expectation

To find the conditional expectation of the random variable X given Y equals y, we can use the definition: \(E[X \mid Y=y] = \int_{-\infty}^{\infty} x \cdot f_{X \mid Y}(x \mid y) dx\) where \(f_{X \mid Y}(x \mid y)\) is the conditional probability density function of X given Y = y.

Use the property of independence

Since X and Y are independent random variables, their joint probability density function can be expressed as the product of their marginal probability density functions: \(f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y)\) As a consequence, the conditional probability density function of X given Y = y is: \(f_{X \mid Y}(x \mid y) = \frac{f_{X,Y}(x,y)}{f_Y(y)} = \frac{f_X(x) \cdot f_Y(y)}{f_Y(y)}\) Since \(f_Y(y) \neq 0\), we can simplify the expression: \(f_{X \mid Y}(x \mid y) = f_X(x)\)

Substitute the conditional PDF

Now that we have the conditional probability density function for independent random variables X and Y, we can substitute it back into the definition of the conditional expectation: \(E[X \mid Y=y] = \int_{-\infty}^{\infty} x \cdot f_{X \mid Y}(x \mid y) dx = \int_{-\infty}^{\infty} x \cdot f_X(x) dx\)

Recognize the expected value of X

The integral in the last expression represents the expected value of the random variable X: \(E[X] = \int_{-\infty}^{\infty} x \cdot f_X(x) dx\)

Make the conclusion

Since the integral in Step 3 is equal to the expected value of X, we can conclude that: \(E[X \mid Y=y] = E[X]\) This holds for all y, as required. Therefore, the expected value of X given Y equals y is the same as the expected value of X for independent random variables X and Y.