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Basic statistics

Basic statistics. Usman Roshan. 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 Probability distributions Central limit theorem.

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Basic statistics

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  1. Basic statistics Usman Roshan

  2. 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 • Probability distributions • Central limit theorem

  3. Random variable • A variable normally takes on different values • Random variable has values with different probabilities • Coin toss example • Dice example • Probabilities must sum to 1

  4. Probability of event • Sample space: set of total possible outcomes • Event space: set of outcomes of interest • Probability 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 counting • Bernoulli trials • Coin tossing example • R function: rbinom

  5. Basic stats • Independent events: coin toss example • Expected value of a random variable – • example of Bernoulli and Binomal • Variance of a random variable • Correlation coefficient (same as Pearson correlation coefficient) • Formulas: • Covariance(X,Y) = E((X-μX)(Y-μY)) • Correlation(X,Y)= Covariance(X,Y)/σXσY • Pearson correlation

  6. Limit theorems • Law 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 numbers • Markov’s inequality • Chebyshev’s inequality • Law of large numbers: sample mean of n i.i.d. random variables Xi converges to true one in probability • Can be proved by applying Chebyshev’s inequality

  8. Limit theorems • Central 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 deviation • How is this useful? Consider modeling the mean height of NJ residents. Can we assume it is normally distributed due to Central Limit Theorem?

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