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Foundations of Probability in R

The geometric distribution

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Foundations of Probability in R

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One of the simplest and most common examples of a random phenomenon is a coin flip: an event that is either "yes" or "no" with some probability. Here you'll learn about the binomial distribution, which describes the behavior of a combination of yes/no trials and how to predict and simulate its behavior.

Exercise 1: Flipping coins in R Exercise 2: Simulating coin flips Exercise 3: Simulating draws from a binomial Exercise 4: Density and cumulative density Exercise 5: Calculating density of a binomial Exercise 6: Calculating cumulative density of a binomial Exercise 7: Varying the number of trials Exercise 8: Expected value and variance Exercise 9: Calculating the expected value Exercise 10: Calculating the variance

In this chapter you'll learn to combine multiple probabilities, such as the probability two events both happen or that at least one happens, and confirm each with random simulations. You'll also learn some of the properties of adding and multiplying random variables.

Exercise 1: Probability of event A and event B Exercise 2: Solving for probability of A and B Exercise 3: Simulating the probability of A and B Exercise 4: Simulating the probability of A, B, and C Exercise 5: Probability of A or B Exercise 6: Solving for probability of A or B Exercise 7: Simulating probability of A or B Exercise 8: Probability either variable is less than or equal to 4 Exercise 9: Multiplying random variables Exercise 10: Expected value of multiplying a random variable Exercise 11: Simulating multiplying a random variable Exercise 12: Variance of a multiplied random variable Exercise 13: Adding two random variables Exercise 14: Solving for the sum of two binomial variables Exercise 15: Simulating adding two binomial variables Exercise 16: Simulating variance of sum of two binomial variables

Bayesian statistics is a mathematically rigorous method for updating your beliefs based on evidence. In this chapter, you'll learn to apply Bayes' theorem to draw conclusions about whether a coin is fair or biased, and back it up with simulations.

Exercise 1: Updating with evidence Exercise 2: Updating Exercise 3: Updating with simulation Exercise 4: Updating after 16 heads Exercise 5: Updating with simulation after 16 heads Exercise 6: Prior probability Exercise 7: Updating with priors Exercise 8: Updating with three coins Exercise 9: Bayes' theorem Exercise 10: Updating with Bayes theorem Exercise 11: Updating for other outcomes Exercise 12: More updating with priors

So far we've been talking about the binomial distribution, but this is one of many probability distributions a random variable can take. In this chapter we'll introduce three more that are related to the binomial: the normal, the Poisson, and the geometric.

Exercise 1: The normal distribution Exercise 2: Approximating a binomial to the normal Exercise 3: Simulating from the binomial and the normal Exercise 4: Comparing the cumulative density of the binomial Exercise 5: Comparing the distributions of the normal and binomial for low n Exercise 6: The Poisson distribution Exercise 7: Approximating a binomial with a Poisson Exercise 8: Simulating from a Poisson and a binomial Exercise 9: Density of the Poisson distribution Exercise 10: Sum of two Poisson variables Exercise 11: The geometric distribution

Exercício atual

Exercise 12: Waiting for first coin flip Exercise 13: Using replicate() for simulation Exercise 14: Simulating from the geometric distribution Exercise 15: Probability of a machine lasting X days Exercise 16: Graphing the probability that a machine still works