L1 - Revision of Probability Theory Flashcards
What is a random experiment?
A random experiment is an experiment whose outcome
cannot be predicted with certainty.
What is a sample space?
The sample space is a set of all possible outcomes of a random experiment
What is an event?
An event is a subset of the sample space. A random event is a subset of the sample space to which we can attach a probability.
What is a random variable?
A random variable is a quantity whose value depends on the outcome of a set of possible random events.
What is a probability distribution?
A random variable has a probability distribution. This specifies the probability that the value of the variable falls within given intervals.
What is A Bernoulli trial?
A Bernouilli trial is special kind of experiment in which the outcome can be classified as either ‘success’ or ‘failure’.
- Also referred to as the Frequentist Approach
What is Relative Frequency?
The relative frequency is the number of successes observed in ‘n’ Bernoulli trials = k/n.
What is the Probability of Success of a Bernoulli trial?
The probability of a success is the value that the relative
frequency converges to as n becomes large.
What is the Cumulative Probability Distribution Function (CDF)?
The cumulative probability distribution function (CDF) gives the probabilities that the random variable takes on values less than or equal to all possible outcomes.
- F(x) in a probability distribution
What is the Frequestist Statistics Approach?
- We assume that the data are generated by some unknown stochastic process with fixed parameters.
- In the classical framework probabilities are thought of as the relative frequency of events when an experiment is repeated many times.
- For the purposes of statistical inference we begin from a position of complete ignorance. The only information available is the data set.
- Classical statistics emphasises the efficient use of data to construct estimates of the unknown parameters of the data generation process.
- Let θ be a parameter of the data generation process. We cannot make probabilistic statements about θ because it is not a random variable. For example, the statement that there is a 95% probability that the parameter lies within a given range makes no sense. Hence we introduce the idea of a confidence interval.
What is the Bayesian Statistic Approach?
- We assume that the data are generated by some unknown stochastic process but the parameters of this process are treated as being random.
- In the Bayesian framework probabilities reflect the subjective degree of belief that given events will occur
- For the purposes of statistical inference we begin with a set of prior beliefs which are updated in response to the information contained in the data set.
- Bayesian statistics emphasises the use of data to update prior beliefs in the most efficient way possible.
- In the Bayesian framework we can make probabilistic statements about the parameters of the data generation process because these are thought of as random variables. We can use the term ‘probability interval’ rather than confidence interval without contradiction.
What is Cross-sectional data?
observe data for different agents
What is Time Series Data?
observe data for one agent at different points in time
