3.1. Probability & Estimation Flashcards
Probability…
Numerical measure of the likelihood of something occuring.
Number of favourable outcomes / total possible outcomes (on a 0-1 scale).
Sample space defines all the possible outcomes of a situation.
Probabilities can be graphically visualised on a tree diagram.
Probability takes two views…
Frequentist:
- The proportion that one specific outcome is obtained in trails evens out as the number of trials approaches infinity.
Subjective:
- Assumes probability is a degree of belief that one holds about the likelihood of something occuring.
Probability distribution…
Details all of the possible outcomes of a probability experiment and the probabilities of each individual outcome.
For example, when flipping a coin, the two possible outcomes are heads and tails, and the probability of each outcome is 1/2.
Random variables…
A random variable is one whose outcome or value is the result of chance and unpredictable.
Discrete: takes on values only at certain points over an interval. This is a specific measurement. For example, flipping a coin three times and counting heads can yield 0, 1, 2 or 3 heads. It can only take on a specific, seperate value with no intermediate value between them.
Continuous: takes on value at every point over an interval. For example, flipping the coin three more times and counting the height it travelled. It can take on an infinite number of values (depending on how strong the flipper was) - it travels through a range.
Statistical inference…
Statistical inference allows us to draw conclusions about a population based on data randomly drawn from a sample of the population.
Before inferring anything about the population however, we must examine the distribution of the random variables within the sample to check it is representative.
Types of probability distributions…
Discrete (binomial and poisson).
Continuous (normal, T-distribution, x2 distribution and F distribution).
Binomial distribution…
The binomial distribution provides a formula for calculating the number of successful trials in a sample.
Check Google Docs for a roulette example at The Venetian Macao.
Properties of normal distribution…
A symmetrical, bell-shaped and unimodal distribution of data.
Ranges from negative to positive infinity.
The area under the curve is equal to 1 and represents the probabilities.
The mean is at the centre of the distribution.
Mean and variance are the parameters.
We can then use Z-scores to calculate probailities.
Check Google Docs for a blackjack example at The Venetian Macao and Grosvenor Casino Aberdeen.
Population distribution theorem…
z=x-μ2n=25,000-17,4009,21650=559.7928684
See Google Docs.
Central limit theorem…
The sample mean, drawn from a population with a mean and variance, has a sampling distribution which approaches a normal distribution as the sample size approaches infinity.
Estimation…
Statistical inference: using sample data to reach conclusions about the population from which the sample was drawn.
Estimator: the rule used to generate an estimate.
The criteria for selecting estimators:
Bias:
- Unbiased estimator gives the correct answer on average.
- Unbiased estimator does not systematically mislead.
- A single use of unbiased estimators does not guarantee a correct estimation.
- The expected value of an unbiased estimator equals the parameter being estimated.
Precision:
- Given two estimators, A and B, A is said to be more precise than B if the estimates it yields are less spread out than those of estimator B.
Interval estimates…
Provides information about how close the point estimate might be to the value of the population parameter.
