# Show that normal distribution has 0 mean and unit variance

For a normal distribution,

$$\phi(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\times e^{\frac{-1(x-\mu)^2}{2\sigma^2}}$$

Question:Prove that the standard normal distribution has 0 mean and unit variance?

Lemmas we will use before we begin:

If $$f(x) = -f(-x)$$ then $$\int_a^a f(x) dx = 0$$ for any $$a \in \mathbb{R}$$

Mean

$$\mathbb{E}[X] = \int_{\mathbb{R}} xf_X(x) dx$$

Using lemma above and noticing that $$xf_X(x)$$ satisfies the property above we are done for zero mean.

Variance

$$\mathbb{V}[X] = \mathbb{E}[X^2] - \mathbb{E}[X]^2 = \mathbb{E}[X^2]$$ as $$\mathbb{E}[X] = 0$$

Hence $$\mathbb{V}[X] = \int_{\mathbb{R}} x^2f_X(x) dx = 1$$