Lecture Notes for ECON3002

Probability

The probability of an outcome is the proportion of the time it occurs in the long run

Notation: For an event $A$ , the number $P r (A)$ indicates the probability that A will occur.

The $Ω$ stands for all possible outcomes.

A discrete random variable i.e the coin (0 or 1 no others)

A continuous random variable i.e the time spent on phone (some times the $f (x)$ could be above 1)

$p (x) = P r (X = x)$

Probability density function

Cumulative distribution function

Uniform distribution have no modes

The basic knowledge: Variance

Prove: if $Y = a + b X$ then $Va r (Y) = b^{2} Va r (X)$ , why?

Since $Va r (X) = E [(x - E (X))^{2}]$ so that $Va r (Y) = E (a + b x - E (a + b x))^{2}$

$Va r (Y) = E [(b (x - \overline{x}))^{2}]$ (the right part is the $Va r (X)$ )

So $Va r (Y) = b^{2} Va r (X)$

Prove: $Va r (X) = E (X^{2}) - [E (X)^{2}]$ :

$Va r (X) = E [(X - E (X))^{2}] = E [X^{2} - 2 XE (X) + E (X)^{2}]$

$= E [X^{2}] - 2 E (XE (X)) + E (X)^{2}]$

==Remember, $E (E (X)) = E (X)$ , so:

$= E [X^{2}] - 2 E (X)^{2} + E (X)^{2}$

$= E [X^{2}] - E (X)^{2}$

The Standard Deviation of $X$ is denoted as $σ_{X}$ , also the $Va r (X)$

The Joint Probability Density Distribution is the probability that two Discrete Random Variable $X$ and $Y$ simultaneously take on particular values. (say $x$ and $y$ )