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Prob & RandVar

Prob & RandVar - Probability and Random Variables 1...

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© 2010 Radha Bose FSU Department of Statistics Probability and Random Variables — 1 We are now going to revisit the idea of "percentage of data". What does the statement "there is a 40% chance of rain tonight" mean? It means that, in the past, whenever weather conditions were pretty much like they are tonight, rain fell in 4 out of 10 instances, i.e., 40% of the time. We can also say there is a 0.4 probability of it raining tonight. Probability is long-run percentage — it is the percentage of times you will see a certain outcome taking place when you look at a series of similar circumstances. TERMS YOU SHOULD UNDERSTAND THE MEANINGS OF #1 and #2 on the next two pages will give you some context for these terms. Random Process — independent trials where the outcome of each trial is unpredictable, but a pattern of outcomes shows up in the long run Sample space (S) — the set of all possible outcomes Probability of an outcome — the long run proportion of times that the outcome happens —the Law of Large Numbers (LLN) assures us that the proportion of occurrences will converge Probability distribution/model — shows sample space along with probability of each outcome (discrete models represented by probability distribution tables or probability histograms) or interval of outcomes (continuous models represented by density curves, probability is area under curve). Equally likely outcomes — have same probability of occurrence, hence a Uniform probability distribution Event — subset of outcomes from the sample space. Events are normally labeled A, B, C, etc ( ) # # of outcomes that contribute to that event P event where outcomes are equally likely total of possible outcomes = Complement of an event — subset consisting of all outcomes in sample space that do not contribute to the event. The complement of event A is written A C or A'. Random variable — a function with numerical values that depend on the outcomes of a random process. Random variables are normally labeled X, Y, Z, etc. Mean, or Expected Value, of a random variable — long run mean, LLN assures us that mean will converge ( ) value Expected X E p x p x p x xp Mean n n X = = + + + = = = ... 2 2 1 1 μ This is actually a weighted mean of the values of the random variable X, where each value is weighted by its probability of occurrence.
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© 2010 Radha Bose FSU Department of Statistics Probability and Random Variables — 2 1. Think of the process of rolling a fair die repeatedly and recording the number that shows up each time. Each roll of the die is considered a __________. The outcome of a roll does not influence the outcome of any other roll, so the rolls are ____________________. Each time we roll the die we never know which number will show up, but in the long run we will see a pattern of outcomes. This fact, together with the fact that we have independent rolls, allows us to describe this process as a _______________ _______________.
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Prob & RandVar - Probability and Random Variables 1...

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