Variance 11/17

For a continuous random variable defined with a density f, and expectation $\mu$, its variance is:

$$Var[X]=E((X-\mu)^2)=\int_{-\infty}^{\infty} (x-\mu)^2 f(x) dx$$

Proposition 1:

Var(X+b)=Var(X)

Proposition 2:

Var(aX)=a² Var(X)

Proposition 3:
Only for independent random variables do we have

$$If\;\; X\; and \; Y\; independent\; Var(X+Y)=Var(X) + Var(Y)$$

Examples:

1.

For the exponential random variable with parameter $\lambda$, I showed:

$$var(X)=\frac{1}{\lambda^2}$$

2.

For U a random uniform on [0,1], I showed: $var(U)=\frac{1}{12}$.

3.

$$Z \sim Normal(0,1) Var(Z)=1$$

For the Normal $(\mu,\sigma^2)$ random variable X:

$$Var(X)=\sigma^2$$

This is the meaning of the second parameter in the definiton of the density, its square root is called the standard deviation and gives the width of the curve.

I explained in class what standardizing a variable meant in general, not just for Normals.