Sample Space 09/24

The set of all possible outcomes is called the sample space and noted $\Omega$.

 

An event is a subset of the sample space $\Omega$.  

Distribution Function We defined the distribution function for a finite sample space to be a positive real valued function m that is defined on $\Omega$ and such that:

$$\sum_{\omega_i \in \Omega} m(\omega_i)=1$$

When the sample space is either finite or countably infinite we say the distribution is discrete.

To an experiment we associate a number, we either number the outcomes, or we assign winnings to them. This number is called a random variable in fact it is a function from the outcomes into the reals.

Random Variable:   We call the numerical outcome of a chance experiment a random variable, and we denote it by X.

Example 1:
When rolling a dice the sample space is

$$\Omega=\{1,2,3,4,5,6\}$$

$$E=\{2,4,6\} \mbox{ is the even number event}$$

Suppose we define the following 3 events: $A=\{1,2,3\}$, $B=\{4,5\}$, $C=\{6\}$ We want to define the probability of these events. If we didn't know any rules but wanted to use the computer to simulate throwing a die and counting how frequent each event was.

Here is the matlab simulation we did in class: