Kurtosis of Normal Distribution I have a basic question about kurtosis and specific distributions. > **Definition:** Let $Y$ be a random variable. $\mu_Y$ and $\sigma_Y$ denote $Y$'s mean and standard deviation respectively. Then, the kurtosis of the distribution of $Y$ is $\frac{E[(Y-\mu_Y)^4]}{\sigma_Y^4}$. My textbook then says "the kurtosis of a normally distributed random variable is $3$." I am wondering whether only standard normal distribution has a kurtosis being 3, or any normal distribution has the same kurtosis, namely $3$. How can all normal distributions have the same kurtosis when standard deviations may vary?

Let $X=\frac{Y-\mu_Y}{\sigma_Y}$. Then $E(\frac{(Y-\mu_Y)^4}{\sigma_Y^4})=E(X^4)=3$, since $X$ is standard normal.