Notation/Terminology:

Random Sample

\[X_1,X_2,...,X_n\]
  • Variables before they are sampled, observed, and “locked in”.
  • They are assumed to be independent and identically distributed (iid).
\[\text{Random Sample} = \text{iid}\]

More Notation:

Suppose that $X_1,X_2,…,X_n$ is a random sample from the normal distribution with mean $\mu$ and variance $\sigma^2$.

We write

\[X_1,X_2,...,X_n\stackrel{iid}{\sim}N(\mu,\sigma^2)\\ E[X_i]=\mu\qquad Var[X_i] = \sigma^2\]

The pdf:

\[f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2\sigma^2}(x-\mu)^2}\]

The expected value:

\[\begin{align} \mu=E[X_i]&=\int_{-\infty}^{\infty}xf(x)dx\\ E[X_i^2]&=\int_{-\infty}^{\infty}x^2f(x)dx \end{align}\]

The variance:

\[\begin{align} \sigma^2&=Var[X_i]=E[(X_i-\mu)^2]\\ &=E[X_i^2] - (E[X_i])^2 \end{align}\]

Any linear combination of normal random variables has, again, a normal distribution.

\[\begin{align} &a_1X_1+a_2X_2+...+a_nX_n\sim N(?,?)\\ &E\bigg[\sum_{i=1}^{n}a_iX_i\bigg]=\sum_{i=1}^{n}a_i\underbrace{E[X_i]}_{\mu}=\mu\sum_{i=1}^{n}a_i\\ &Var\bigg[\sum_{i=1}^{n}a_iX_i\bigg]\stackrel{indep}{=}\sum_{i=1}^{n}a_i^2\underbrace{Var[X_i]}_{\sigma^2}=\sigma^2\sum_{i=1}^{n}a_i^2 \end{align}\]

In particular, if

\[X_1,X_2,...,X_n\stackrel{iid}{\sim}N(\mu,\sigma^2)\]

Then

\[\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_i\sim N\bigg(\mu,\frac{\sigma^2}{n}\bigg)\]

Note that the bigger the sample, the less variability we’re going to see.

The $N(0,1)$ distribution is known as the standard normal distribution.

We typically use the letter $Z$:

\[Z\sim N(0,1)\]

The cumulative distribution function (cdf)

\[\begin{align} \phi(z) &= P(Z\le z)\\ &=\int_{-\infty}^{z}\frac{1}{2\pi}e^{-\frac{1}{2}x^2}dx \end{align}\]

png

  • $X\sim N(\mu,\sigma^2)\implies\frac{X-\mu}{\sigma}\sim N(0,1)$
  • $Z\sim N(0,1)\implies\sigma Z+\mu\sim N(\mu,\sigma^2)$
Example 1

Let $X\sim N(2,3)$.

Then

\[\begin{align} P(X\le4.1)&=P\Bigg(\frac{X-\mu}{\sigma}\le\frac{4.1-2}{\sqrt{3}}\Bigg)\\ &=P(Z\le1.21)\\ \approx0.8868 \end{align}\]

We can use R: pnorm(1.21)

Example 2

$X_1,X_2,…,X_10\stackrel{iid}{\sim}N(2,3)$

$\overline{X}\sim N(\mu,\sigma^2/n)=N(2,3/10)$

\[\begin{align} P(\overline{X}\le2.3) &= P\Bigg(\frac{\overline{X}-\mu_{\overline{X}}}{\sigma_\overline{X}}\le\frac{2.3-2}{\sqrt{3/10}}\Bigg)\\ &=P(Z\le0.5477)\\ &\approx0.7081 \end{align}\]

Critical Values

  • Values that cut off specified areas under pdfs.
  • For the N(0,1) distribution, we will use the notation $Z_\alpha$
Example

png

We can use R to compute the critical value: qnorm(0.95)