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:
- Two possible outcomes
Each trial has two possible outcomes. - Same probability of success
Each trial has the same probability of success - 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 
