Resource Economics 212 October 12 2016
Textbook Notes
TBL 6 Random Experiments, Outcomes, and Sample Spaces . Random Experiments o An observational process whose results cannot be known in advance. Imagine the experiment before any attempt to collect data. Outcome o Single observation of the random experiment - The outcome is imagined, because the experiment is a thought experiment. Not what you can collect in reality, but a matter of what is a possible occurrence. Sample Space ttDenoted by S) o Collection of all possible outcomes of the random experiment. All are dreamed up, using some knowledge of the structure of the problem. Random Experiment Example 1 o Random Experiment: tossing two coins oPossible Outcomes: Heads ttH), Tails ttT) 0 Sample Space ttS): ttH,H), ttH,T), ttT,H), ttT,T) Random Experiment Example 2 o Experiment: Rolling a pair of dice o Possible Outcome: tt4,1) O Sample Space ttS): tt1,1), tt1,2), tt1,3)....... ...t6,6) :36 Possible Outcomes Types of Sample Spaces o Discrete or Countable Each element of the sample space can be arranged and counted one at a time Example: Tossing two coins. Types of Countable Sample Spaces : Finite You will eventually count all the elements in the sample space. Countably Infinite o Each element of the sample space can be arranged and counted one at at a time, BUT You will never finish counting the elements of the sample space.
o Uncountable : Possible outcomes cannot be arranged and counted. - EG: observing the final grades ttIn percentages) of students in ResEcon. :S= [0,100]2
Events: Simple and Compound Event: a collection of outcomes in the sample space o Simple Event Contains a single outcome. Denoted by E Example: the toss results in two tails. ttT,T) Compound Event Contains more than one outcome. Denoted by A ttor B, or C,..) Example: the toss results in at most one head. ttH,T), ttT,H), ttT,T) Probability: Definition and Rules . Probability: numerical measure of the likelihood that an event will occur. o Rules: Rule 1: For any event, A, 0 =< PttA) =< 1 Rule 2: If the sample space is finite, then =(1+(2+...=1 P (A) = 0 implies A is impossible P (A) = 1 implies A is certain. Probability of an event cannot be negative.
Assigning Probabilities Relative Frequency ttEmpirical) Approach o Perform an experiment many times ttn), with the experiments repeated identically o Record the outcomes o The probably of an event, E, ios Number of Observations of E, (f)_f : PttE)= Number of Observations(n) n Example: tossing a coin Classical Approach o Probabilities are assigned to events without performing experiments, and are derived from logic or theory. -EG: What is the probability that the roll of a die will result in fi up? Subjective Approach o Probabilities are assigned based on experience or informed judgement. o Expresses degree of belief that an event will occur. o EG: Who will win the Super bowl?
Events and their Complements : Complement
o Consists of every event in the sample space S except f