4.2

4.2.1 Random Variable and its Properties

Given a random variable $X$,

1. Mean

   $$ \mu = \left \{ \begin{array}{lc} \sum xp(x),\ & discrete \\ \int xf(x)dx,\ & continuous \\ \end{array} \right . $$

   where $f(x)$ is the p.d.f, and $p(x)$ is the p.m.f

2. Median

   • A number $m$ such that $P(X\le m)\ge\frac{1}{2}$ and $P(X\ge m)\ge\frac{1}{2}$ for discrete r.v. $X$.

   • The area under the p.d.f $f(x)$ is divided into 2 parts by the vertical line $x=m$, with equal area $\frac{1}{2}$. (continuous)

     

3. Mode

   • The value appears the most time. (discrete)
   • Local maxima of the p.d.f of $X$(could be more than one) (continuous)

4. Variance

   $$ \sigma^2 = \left \{ \begin{array}{cl} \sum(x-\mu)^2p(x),&discrete\\ \int(x-\mu)^2f(x)dx,&continuous \end{array} \right . $$

   where $f(x)$ is the p.d.f, and $p(x)$ is the p.m.f

The cumulative distribution function (c.d.f) for a random variable $X$ is

$$ F(x) = P(X\le x) = \left \{ \begin{array}{cl} \sum_{t\le x}p(t), & discrete\\ \int^x_{-\infin}f(t)dt, & continuous \end{array} \right . $$

where $f(t)$ is the p.d.f, and $p(t)$ is the p.m.f

4.2.2 Sampling Distributions

Definition:

Let $X_1,X_2,\cdots,X_n$ be $n$ independent random variables, eadch having the p.d.f $f(x)$. Define $X_1,X_2,\cdots,X_n$ to be a random sample of size $n$ from the population. Its joint p.d.f is $f(x_1,x_2,\cdots,x_n)=f(x_1)f(x_2)\cdots f(x_n)$.

Definition:

The probability distribution of a statistic is called a sampling distribution.

Why are we so interested in them? They allow us to make inferences on the unknown population parameters, e.g. $\mu$ and $\sigma^2$.

4.2.3 What is the Sampling Distribution of $\bar X$?

The sampling distribution of $\bar X$ with sample size $n$ is the distribution that results when an experiment is conducted over and over (always with sample size $n$) and many values of $\bar X$ result.