Tag: probability
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Limit theorems explained
Before we dive into the theorems let’s tackle a concept one often sees in statistics: the notion of independent, identically distributed (iid) random variables. Whether we’re drawing a sample from a population or conducting a series of experiments like coin flips, we can assess whether iid holds true or not as follows: Independent? Here we…
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Poisson vs Exponential distributions
These distributions are related yet different – here’s a comparison that hopefully clears up any confusions! Poisson Exponential Number of events that occur in an interval of time Time taken between 2 events occurring For example… the number of Metrorail trains that arrive at the platform in an hour For example… the time between one…
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Law of total probability – worked examples
According to Wikipedia the law of total probability “expresses the total probability of an outcome which can be realized via several distinct events”. We can also think of this as the marginal probability: irrespsective of what road we took to get to this outcome, what is the total likelihood of the outcome occurring? Example 1…
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Expected value refresher
The expected value of an event is its most likely outcome. Assign each potential result a probability. The expected value is sum of all the potential results x their respective probabilities: ∑ (potential_result1 x probability1,… potential_resultn x probabilityn) Consider the simplest example possible, the coin flip. You’ll be paid R10 if you pick tails, but…
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Co-variance, Correlation & Linear Regression
Typically we have 2 sets of values and we want to find out if these 2 sets of values are related, and if so how, and by how much? Could height be indicative of weight? Could hours of practice be related to how many errors are made in a mathematical test paper? Co-variance is a…
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Conditional probability refresher
Probability is always… A number between 1 (certain) and 0 (impossible): The probability experiment This is process that can be repeated and in which the outcome will be uncertain, for example rolling a dice, tossing a coin, or selecting a card from a pack. The sample space This is the set of all possible outcomes…
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T testing – a worked example
A simple one-sample T-test This variant on hypothesis testing is used when you have limitations, specifically: The population standard deviation (σ) is unknown and your sample size (n) is <30 The fundamentals The formula is a variant of what we’ve seen thus far, where x̄ = your sample mean, μ = a hypothesized population mean,…
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Proportion testing
Using everything we’ve learned so far about the central limit theorem, the z-score, and hypothesis testing, we can now also perform proportion testing! There are just a few new concepts to add into the mix: The preliminary terrors – notation & terminology p = the proportion of items that falls into H0 q = the…
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Hypothesis testing basics
A simple example of hypothesis testing is where we know what “normal” is, and we want to evaluate whether some sample conforms to our understanding of “normal”, or is so unusual that it’s indicative of an actual shift in behaviour or pattern. Make your hypothesis statement If I…(do this to an independent variable)….then (this will…
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Populations and samples
Populations We can think of the population as the complete set of “things”, whatever the “things” are that are under consideration – for example if we’re interested in studying the height of men in South Africa, then the population would be all adult men in South Africa. A population can be described by parameters. Here’s…
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