Quick Recapitulation

This chapter introduced random variables and the main probability distributions used in applied statistics. Its key ideas and formulae are gathered below.

Random variables and probability distributions

1. A random variable $X$ is a real-valued function whose domain is the sample space of a random experiment. More than one random variable can be defined on the same sample space.
2. A discrete random variable takes distinct, countable values, while a continuous random variable is obtained by measurement and can take any value between two given values.
3. A probability distribution table lists every possible value of $X$ together with its probability; because these values exhaust the sample space, the probabilities always satisfy

Expectation and variance

4. For $X$ taking values $x_1,x_2,\dots,x_n$ with probabilities $p_1,p_2,\dots,p_n$, the mathematical expectation (the weighted average of the values) is

The mean expectation is a parameter, not a statistic.

5. The variance measures how much $X$ varies about its mean $\mu=E(X)$:

The standard deviation is $\sigma=\sqrt{\operatorname{Var}(X)}$.

Bernoulli trials and the binomial distribution

6. A collection of trials is a set of Bernoulli trials if the number of trials is finite, the trials are independent, each trial has exactly two outcomes (success and failure), and the probability of success stays the same in every trial.
7. The probability of $r$ successes in $n$ Bernoulli trials is

where $p$ is the probability of success, $q$ that of failure, $p+q=1$, and $r=0,1,2,\dots,n$. This is the $(r+1)$th term in the binomial expansion of $(q+p)^{n}$, and the distribution is denoted $B(n,p)$.

8. For a binomial distribution, $\text{Mean}=np$, $\operatorname{Var}=npq$ and $\text{Standard Deviation}=\sqrt{npq}$.

Poisson distribution

9. If the discrete random variable $X$ counts the number of occurrences of an event over a period of time and follows a Poisson distribution, then

where $e=2.71828\dots$ is Euler's number and $\lambda=E(X)=\operatorname{Var}(X)$ is a positive real number.

Normal distribution

10. A continuous random variable $X$ is defined through its probability density function (PDF). $X$ follows a normal (Gaussian) distribution with constant parameters $\mu=E(X)$ and $\operatorname{Var}(X)=\sigma^{2}$, written $X\sim N(\mu,\sigma^{2})$, with density

where $\mu$ is the mean and $\sigma$ the standard deviation.

11. Key features of a normal distribution: the mean, median and mode coincide; the bell-shaped curve has a single peak (it is unimodal); it is symmetric about $x=\mu$, so half the values fall below the mean and half above; the total area under the curve is $1$; and it is completely described by the two values $\mu$ and $\sigma$.

Standard normal distribution, Z-score and Z-test

12. When $\mu=0$ and $\sigma=1$ the distribution is the standard normal distribution. Any normal value is converted to a Z-score by

which is positive when the data point lies above the mean and negative when it lies below. …