How to Derive Moments of a Normal Distribution Using Differentiation Under the Integral Sign
In this article, we will explore the process of deriving moments of a normal distribution using differentiation under the integral sign. This technique is particularly useful in probability theory and statistics.
Introduction to Differentiation Under the Integral Sign
Differentiation under the integral sign, also known as Feynman's trick, is a technique used to evaluate integrals that involve a parameter. The basic idea is to differentiate the integrand with respect to this parameter, which can simplify the integral and make it more manageable.
Applying Differentiation Under the Integral Sign to Derive Moments
Consider a normally distributed random variable μ2r with mean 0 and variance σ2. We aim to derive the rth moment, denoted as μ2r, using this technique.
Step 1: Express the rth Central Moment
The rth central moment of a normally distributed random variable can be written as:
[ mu_{2r} int_{-infty}^{infty} x^{2r} frac{1}{sigma sqrt{2pi}} e^{-frac{x^2}{2sigma^2}} , dx ]
Step 2: Differentiate with Respect to σ
To find the derivative of μ2r with respect to σ, we differentiate the above integral with respect to σ while treating x as a constant:
[ frac{dmu_{2r}}{dsigma} -int_{-infty}^{infty} frac{1}{sigma^2 sqrt{2pi}} x^{2r} e^{-frac{x^2}{2sigma^2}} , dx ]
Next, we use the fact that the integral of a Gaussian function can be differentiated under the integral sign:
[ frac{dmu_{2r}}{dsigma} -int_{-infty}^{infty} frac{1}{sigma^2 sqrt{2pi}} x^{2r} e^{-frac{x^2}{2sigma^2}} , dx ]
Step 3: Multiply by σ3
Now, we multiply both sides of the equation by σ3 to simplify the expression:
[ sigma^3 frac{dmu_{2r}}{dsigma} -sigma^3 int_{-infty}^{infty} frac{1}{sigma^2 sqrt{2pi}} x^{2r} e^{-frac{x^2}{2sigma^2}} , dx ]
Simplify the left side and the integral:
[ sigma^3 frac{dmu_{2r}}{dsigma} -int_{-infty}^{infty} frac{sigma x^{2r}}{sqrt{2pi}} e^{-frac{x^2}{2sigma^2}} , dx ]
Step 4: Apply Differentiation Again
We now differentiate the right-hand side of the equation with respect to σ again to obtain:
[ frac{d^2mu_{2r}}{dsigma^2} -int_{-infty}^{infty} frac{1}{sigma^4 sqrt{2pi}} x^{2r} e^{-frac{x^2}{2sigma^2}} , dx ]
Next, we multiply both sides by σ4 to simplify:
[ sigma^4 frac{d^2mu_{2r}}{dsigma^2} -int_{-infty}^{infty} frac{1}{sqrt{2pi}} x^{2r} e^{-frac{x^2}{2sigma^2}} , dx ]
Define the moment μ2r2 as:
[ mu_{2r2} int_{-infty}^{infty} frac{1}{sqrt{2pi}} x^{2r2} e^{-frac{x^2}{2sigma^2}} , dx ]
Step 5: Rearrange and Simplify
After simplifying the terms, we obtain:
[ sigma^4 frac{d^2mu_{2r}}{dsigma^2} -sigma mu_{2r2} ]
Rearrange the terms to isolate μ2r2:
[ mu_{2r2} frac{d^2mu_{2r}}{dsigma^2} / (-sigma) ]
Thus, we have successfully derived the relationship between the moments of a normally distributed random variable using differentiation under the integral sign.
Conclusion
Through the application of differentiation under the integral sign, we have derived the relationship between moments of a normally distributed random variable. This technique offers a powerful tool in the realm of probability theory and statistics, enabling us to derive moments and understand the properties of random variables more deeply.