Disclaimer

The key points summarized here are derived from Yeh (2012)¹ and Hogg, Tanis, & Zimmerman (2013)². These notes are not entirely original but are recorded and organized for personal study and better understanding. If you find them useful, please cite the original works, and always refer to the original texts for the most accurate information.

Random Experiment

The study of probability originates from experiments with uncertain outcomes, referred to as random experiments.

1. A random experiment is an experiment whose outcome cannot be predicted in advance.
2. A random experiment can be repeated under identical conditions.
3. The probability of a random experiment can be given by the number of favourable outcomes / total number of outcomes.

Sample Space

The sample space is the set of all possible outcomes of a random experiment.

It is denoted by $S$, and it is the universe of all possible outcomes.

Sample Point

A sample point is a specific outcome of a random experiment. It is denoted by $x$, and it is a member of the sample space $S$.

Examples

1. Consider a random experiment in which a fair coin is tossed repeatedly until the first occurrence of a heads ($H$). Let $H$ represent heads and $T$ represent tails. Then, the sample space $S$ is given by:

   $$S = \{H, TH, TTH, ...\}$$

   where each outcome $x \in S$ corresponds to a finite sequence of tosses ending with the first occurrence of heads. Formally, each sample point can be represented as:

   $$x = T^{k-1}H, \quad k \in \mathbb{N}$$

   where $k$ denotes the trial number on which the first heads occurs, and $T^{k-1}$ represents $k-1$ consecutive tails before the heads.

2. Consider a random experiment in which ramdomly sample the lifespan of light bulbs in a factory. Let $t$ represent the lifespan(in hours) of a bulb. Then, the sample space $S$ is given by:

   $$S = \{t: t \ge 0\}$$

Based on the different properties of sample points within the sample space, the sample space can be classified into discrete sample space and continuous sample space.

Discrete Sample Space

A sample space is called a discrete sample space if all its sample points can be put into a one-to-one correspondence with the set of positive integers($\mathbb{N}$).

Continuous Sample Space

A sample space is called a continuous sample space if all its sample points form a continuous set, such as an interval or a region in the Euclidean space $R^{k}$, where $𝑘$ represents the dimensionality of the region.

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Event

A subset of a sample space whose elements share a common property is called an event.

Elementary Event

For any sample point $x$ in the sample space $S$ the set ${x}$ is called an elementary event.

Occurrence of an Event

Given a random experiment where an outcome $x$ is observed, and an event $A \subseteq S$, if $x \in A$, then event $𝐴$ is said to have occurred.

Sure Event(Certain Event)

The event that always occurs. In probability theory, this is simply the sample space $S$ itself, which includes every possible outcome of a random experiment.

Null Event

Let $\varnothing$ be an element of $S$. Since $\varnothing$ contains no sample points from $S$, it is called the null event.

Example

Toss a fair coin 3 times. Let $H$ represent heads and $T$ represent tails. Then, the sample space $S$ is given by:

$$S = \{HHH, HHT, HTT, HTH, THH, TTH, THT, TTT\}$$

Let

$$A_{i}=\{\text{ i heads occurred during the 3 tosses }\} \text{, where } i=0,1,2,3$$

Then

$$\begin{align*} A_{0} &= \{TTT\} \\ A_{1} &= \{HTT, TTH, THT\} \\ A_{2} &= \{THH, HTH, HHT\} \\ A_{3} &= \{HHH\} \end{align*}$$

According to the above definition,

• All $A_{i}$ are events.
• $A_{0}$, $A_{3}$ are elementary events.
• If $\{HTT\}$ happenned, then is so called $A_{1}$ happened, $A_{0}$, $A_{2}$ and $A_{3}$ not happened.

Corrections Are Always Welcome

If you find any mistakes or have suggestions for improvement, please feel free to submit a pull request. Your corrections and contributions will be incorporated into these notes. Thank you for your time!

References

1. Yeh, H.-C. (2012). Advanced statistics (8th ed.). College of Law National Taiwan University Library and Stationery Department.

2. Hogg, R. V., Tanis, E. A., & Zimmerman, D. (2013). Probability and statistical inference (9th ed.). Upper Saddle River, NJ: Pearson.