learn-co-students/dsc-2-19-03-statistical-distributions-online-ds-pt-011419

Statistical Distributions

Introduction

In this lesson, we'll discuss the concept of statistical distributions, and review prior the prior distributions we've covered--the Normal Distribution and the Binomial Distribution.

Objectives

You will be able to:

• Understand and explain why statistical distributions are useful to data scientists
• Identify real-world use cases for problem solving with statistical distributions
• Review prior knowledge on Uniform, Normal, and Binomial Distributions

Review: What is a Statistical Distribution?

In this section, we'll be learning about some more advanced statistical distributions, including:

1. Negative Binomial Distribution
2. Geometric Distribution
3. Poisson Distribution
4. Exponential Distribution

To make learning these distributions easier, it will be helpful to review what we have alreadylearned about of the more commom/more basic distributions such as the:

1. Uniform Distribution
2. Binomial Distribution
3. Normal Distribution

Although all of these distributions are quite different, in a way, they quite the same in how we use them--these statistical distributions tell us about the distribution of probability of an event or condition given a set of parameters. Recall that distributions can be either discrete or continuous. Let's review what those terms mean.

Continuous Distributions

The event we're measuring can be any real number. We can get infinitely more precise if we choose to, and fractions/decimals are okay. A good example of a continuous distribution is height. A person's height can be a value like 5' 9", or 5' 10", and it can also be literally any number in between those, up the level of precision we choose.

Discrete Distributions

Discrete Distributions deal with a series of boolean events or trials. These events are often called Bernoulli Trials. Recall that Bernoulli Trials always have a binary outcome, and fractions/decimals are not allowed--there is no such thing as half a trial. A Good example of a discrete distribution is a coin flip. The coin either lands on heads or it doesn't, and there is no such thing as a half-flipped coin. Note that we can also represent events with multicategorical outcomes as a discrete distribution, as long as we determine a single outcome of the event to be a success, and all other possible outcomes as a failure. For instance, we can describe the event of rolling a six-sided die as a Bernoulli event if we classify rolling a 3 a success, and any other number a failure.

Now, let's quickly review the previous distributions we've already learned about.

Review: Uniform Distribution

The Uniform Distribution describes an event where every possible outcome is equally likely. No single outcome carries any more or less probability of happening than any other possible outcome. The Uniform Distribution can be discrete or continuous.

A Discrete Uniform Example would be rolling an $n$-sided die. No matter how many sides the die has, a person is equally likely to roll every side.

A Discrete Continuous Example would be the waiting for an elevator that is equally likely to be at any floor in the building when you call it, and can take between 0 and 40 seconds to arrive at your floor. Since the elevator is equally likely to at any given floor, we can assume every amount of time between 0 and 40 seconds (decimals and fractions allowed, to an infinite amount of precision) is equally likely to be the amount of time it takes for the elevator to get to us.

Uniform Distribution Mean:

$$\frac{b + a}{2}$$

Uniform Distribution Standard Deviation:

$$\sqrt{\frac{(b - a)^2}{12}}$$

NOTE: If you're confused why there is a 12 in the denominator of the formula Standard Deviation for a Uniform Distribution, you're not alone. The short answer is that it involves calculus. As a data scientist, you don't need to understand the derivation of this formula and where this 12 comes from--however, if you're interested, this quora answer gives an excellent explanation!

Review: Normal Distribution

The Normal Distribution is also called a Bell Curve, and is the distribution that people most often have some familiarity with. The normal distribution is for continous-valued distributions.

An example of is human height.

The normal distribution is one of the most well-studied distributions. Statisticans and Data Scientists are usually very happy when they find the data they are working with is normally distributed, because we can immediately answer a lot of questions about it. For instance, we know that:

• 68% of probability mass falls within 1 standard devation of the mean
• 95% of probability mass falls within 2 standard devation of the mean
• 99.7% of probability mass falls within 3 standard devation of the mean

Data Scientists often normalize continuous-valued data by converting to the Standard Normal Distribution, also known as the Z-distribution. Recall that a Z-score tells us how many standard deviations away from the mean a given value is. We can convert any value to it's corresponding Z-score by subtracting the mean of the dataset and dividing by the standard deviation. If we do this for every value in a dataset, then we have a normalized dataset consisting of Z-scores. Recall that values smaller than the sample mean will have a negative z-score, the mean will have a Z-score of 0, and values greater than the sample mean will have a positive Z-score.

The formula for the Probability Density of the Normal Distribution is:

$$f(x\ |\ \mu,\ \sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^2}}e$$

Where:

$x$ is the point we want to calculate the probability for

$\mu$ is the mean of the sample

$\pi$ is a mathematical constant, the irrational number $3.14159$

$\sigma^2$ is the variance (since $\sigma$ is the standard deviation)

$e$ is Euler's Constant, also known as the Base of the Natural Logarithm, $2.71828$

An example question we can answer with the Normal Distribution is "what percentage of people are at least 2 inches shorter than the global average hieght?"

Review: Binomial Distribution

The Binomial Distribution is the discrete version of the normal distribution. This distribution describes the probability distribution for a of a given number of successes in a set of repeated Benoulli Trials, also known as a Binomial Experiment. Recall that Binomial Experiments have the following constraints:

• Each experiment consists of $n$ repeated trials.

• The outcome of each trial is binary, resulting in either success or failure (it doesn't matter which outcome we label as success or failure, just that we're able to assign the labels).

• The probability $p$ of a given outcome is the same on every trial.

• The trials are independent. The results of a given trial are not influenced by prior trial results, and will not influence future trial results in turn.

We can calculate the mean of the binomial distribution with the following formula:

$$\mu = n * p$$

where $n$ is the number of trials, and $p$ is the probability of success for a given trial.

The Standard Deviation for a Binomial Distribution is:

$$\sigma = \sqrt{n * p * (1 - p)}$$

The formula for the Point Probability of the Binomial Distribution is:

$$ \Big(\frac{n!} {x! (n-x)!}\Big) p^x (1 - p)^{n - x}$$

where $n$ is the number of trials, $p$ is the probability of success for a given trial, and $x$ is the number of successes.

The formula for the Cumulative Probability of the Binomial Distribution is:

$$\sum_{i=0}^{x} \Big(\frac{n!} {x! (n-x)!}\Big) p^x (1 - p)^{n - x} $$

where $n$ is the number of trials, $p$ is the probability of success for a given trial, and $x$ is the number of successes.

NOTE: The above equation may look scary at first glance, but it's something that you can easily figure out if we describe it with code instead of mathematical notation. This formula represents the combined probabilities of every possible value less than and up to our value $x$. In python, this simply means writing a for loop to help us total the Point Probability for every value starting at $0$ and going to x, inclusive!

An example question we could answer with the Binomial Distribution is "if I flip a fair coin 5 times, what is the probability that exactly 2 of those flips lands on heads?"

Summary

In this lesson, we reviewed the Uniform, Normal, and Binomial Distributions, as well as what it means to say if a distribution is Discrete or Continuous.