Statistics, the art and science of learning from data by A. Agresti (fourth edition) – Chapter 6 summary

SUMMARIZING POSSIBLE OUTCOMES AND THEIR PROBABILITIES
All possible outcomes and probabilities are summarized in a probability distribution. There is a normal and a binomial distribution. A random variable is a numerical measurement of the outcome of a random phenomenon. The probability distribution of a discrete random variable assigns a probability to each possible value. Numerical summaries of the population are called parameters and a population distribution is a type of probability distribution, one that applies for selecting a subject at random from a population.

The formula for the mean of a probability distribution for a discrete random variable is:

μ= ΣxP(x)

It is also called a weighted average, because some outcomes are likelier to occur than others, so a regular mean would be insufficient here. The mean of a probability distribution of random variable X is also called the expected value of X. The standard deviation of a probability distribution measures the variability from the mean. It describes how far values of the random variable fall, on the average, from the expected value of the distribution. A continuous variable is measured in a discrete manner, because of rounding. A probability distribution for a continuous random variable is used to approximate the probability distribution for the possible rounded values.

PROBABILITIES FOR BELL-SHAPED DISTRIBUTIONS

The z-score for a value x of a random variable is the number of standard deviations that x falls from the mean. It is calculated as:

The standard normal distribution is the normal distribution with mean  and standard deviation  . It is the distribution of normal z-scores.

PROBABILITIES WHEN EACH OBSERVATION HAS TWO POSSIBLE OUTCOMES
An observation is binary if it has one of two possible outcomes (e.g: accept or decline, yes or no). A random variable X that counts the number of observations of a particular type has a probability distribution called the binomial distribution. There are a few conditions for a binomial distribution:

1. Two possible outcomes
   Each trial has two possible outcomes.
2. Same probability of success
   Each trial has the same probability of success
3. Trials are independent

The formula for the binomial probabilities for any n is:

The binomial distribution is valid if the sample size is less than 10% of the population. There are a couple of formulas for the binomial distribution:

 and