Basic probability and stats Random variable Probability of an event Coin toss example Independent random variables Mean and variance of a random variable Correlation between random variables ID: 1027455
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1. Basic statisticsUsman Roshan
2. Basic probability and statsRandom variableProbability of an eventCoin toss exampleIndependent random variablesMean and variance of a random variableCorrelation between random variablesProbability distributionsCentral limit theorem
3. Random variableA variable normally takes on different valuesRandom variable has values with different probabilitiesCoin toss exampleDice exampleProbabilities must sum to 1
4. Probability of eventSample space: set of total possible outcomesEvent space: set of outcomes of interestProbability of an event is (size of event space)/(size of sample space)Counting: how many ways to pick k unique items from a set of n items?Probability and countingBernoulli trialsCoin tossing exampleR function: rbinom
5. Basic statsIndependent events: coin toss exampleExpected value of a random variable – example of Bernoulli and BinomalVariance of a random variableCorrelation coefficient (same as Pearson correlation coefficient)Formulas:Covariance(X,Y) = E((X-μX)(Y-μY))Correlation(X,Y)= Covariance(X,Y)/σXσYPearson correlation
6. Limit theoremsLaw of large numbers: empirical mean converges to true mean as we do more trials (follows from Chebyshev’s and Markov’s inequalities)
7. Law of large numbersMarkov’s inequalityChebyshev’s inequalityLaw of large numbers: sample mean of n i.i.d. random variables Xi converges to true one in probabilityCan be proved by applying Chebyshev’s inequality
8. Limit theoremsCentral limit theorem: average of sampling distribution converges to a normal distribution as we do more trials. Specifically, it is normally distributed with mean equal to the true mean μ and standard deviation equal to σ/sqrt(n) where n is number of trials and σ is true standard deviationHow is this useful? Consider modeling the mean height of NJ residents. Can we assume it is normally distributed due to Central Limit Theorem?