Completing the Square: How do we Derive the Expectation of a Lognormal Variable?

[rwinston]

Hi all

A basic question here:

I am deriving the equation for the expectation of a lognormal variable. This involves rearranging the contents of the integral

$\int_{-\infty}^{+\infty} e^x e^{-(x-\mu)^2/2\sigma^2}dx$

A proof I have seen completes the square like so:

$x-\frac{(x-\mu)^2}{2\sigma^2} = \frac{2\sigma^2x-(x-\mu)^2}{2\sigma^2}$

$= \frac{(x-(\mu+\sigma^2))^2}{2\sigma^2} + \mu + \frac{\sigma^2}{2}$

So, trying this (ignoring the 2\*sigma^2 denominator for now):

$2\sigma^2x-(x-\mu)^2 = 2\sigma^2x-(x^2-2\mu x +\mu^2)$

$=-x^2+(2\mu+2\sigma^2)x-\mu^2$

$\Rightarrow x^2-(\mu+2\sigma^2)x =\mu^2$

$x^2-(2\mu+2\sigma^2)x+(-\mu-\sigma^2)^2 = -\mu^2 + (-\mu-\sigma^2)^2$

When attempt to express the LHS as a square:

$(x-(\mu+\sigma^2))^2 = -\mu^2 + (\mu^2 + 2\sigma^2\mu + \sigma^4)$

$\Rightarrow (x-(\mu+\sigma^2))^2 = 2\sigma^2\mu+\sigma^4)$

Bringing over the RHS terms, and factoring in the 2*sigma^2 denominator:

$\Rightarrow \frac{(x-(\mu+\sigma^2))^2}{2\sigma^2} + \mu + \frac{\sigma^2}{2} = 0$

[ok , got it now. thanks to the poster below]

 

[arildno]

1st line mistake: When expanding the square parenthesis into the full parenthesis, there shall be a + in front of $\mu^{2}$, not - as you have done.

 

[tacman]

Completing the square is a technique used to simplify quadratic expressions and make them easier to work with. In this case, we are trying to derive the expectation of a lognormal variable, which can be represented by the integral shown above. By completing the square, we can simplify the expression and make it easier to integrate.

The process of completing the square involves adding and subtracting a constant term to the expression in order to create a perfect square. In this case, we are adding and subtracting (\mu + \sigma^2)^2 to the expression. This allows us to express the quadratic term as a perfect square, which makes it easier to integrate.

By rearranging the contents of the integral using this technique, we are able to express the integral as a Gaussian integral, which is much easier to solve. This allows us to derive the expectation of a lognormal variable by solving the Gaussian integral and evaluating the resulting expression.

In summary, completing the square is a useful technique for simplifying quadratic expressions and making them easier to work with. In the case of deriving the expectation of a lognormal variable, it allows us to express the integral as a Gaussian integral, which can be solved to obtain the desired result.