Probability+Distributions

toc = Definition = A probability distribution is a table or an equation that links an event with the probability of its outcome. There are two types of probability distributions: discrete distribution and a continuous distribution. Discrete distributions have a limited number of possible outcomes while continuous distributions have an infinite number of possible outcomes.

Before We Start
Here are the some symbols to be familiar with:
 * //x//: Number of successes.
 * //n//: Number of trials performed successfully.
 * //P//: Probability of success on an individual trial.
 * //Q//: Probability of failure on an individual trial.
 * µ: The Mean (Mu - like the Pokemon)

Discrete Distribution
Discrete probability distribution has a limited, countable number of possible outcomes. For example, if you were to create a discrete probability distribution of the possible outcomes of drawing coins from a bag, while your expected return might be 8.5 cents, it is impossible to predict drawing an 8.5 cent coin in your probability distribution; coins only come in 1, 5, 10, and 25 cent denominations.

Here is an example of a discrete probability distribution: Based on information gathered from his previous trips, this is the probability distribution of the expected number of books Socrates will buy at the bookstore. Although the expected value for any one trip to the book store is that Socrates will buy 1.3 books this is impossible as he can only purchase an integer number of books. This is why 1.3 is not included in the probability distribution.
 * Number of books || 0 || 1 || 2 || 3 ||
 * Probability || .2 || .4 || .3 || .1 ||

Here is a graph of a discrete probability distribution (note: not the one in the above example): -In this graph, only discrete numbers are used. It represents the probability distribution of rolling any of the following numbers using two dice.

Binomial Distribution:
Binomial Distribution is one example of Discrete Distribution

Binomial Distribution the following properties: 1. Must have repeated trials. 2. All trials must be independent (eg: replacing a ball in a jar) 3. The probability of success must be the same for each trial. (eg: same number of balls in the jar every time)

Example 2: Here's an example of binomial distribution using repeated coin tosses (you can flip the coin as well, it will still work) You toss three coins. Let x be the number of times the coin lands heads side up. Here is a probability distribution for x and with its mean and standard deviation.

Probability Distribution: Mean: (represented by the greek symbol µ )
 * x || P(x) ||
 * 0 || 1/8 ||
 * 1 || 3/8 ||
 * 2 || 3/8 ||
 * 3 || 1/8 ||

µ 0*1/8 + 1*3/8 + 2*3/8+ 3*1/8 1.5 µ = 1.5

Standard Deviation: Let ø represent the standard deviation. **ø** = Square Root [ ∑ (x^2 *P(x)) - µ^2 ] ∑ : the mathematical sum of = sqrt [ (0 * 1/8) + (1 * 3/8) + (2^2 * 3/8) + (3^2 * 1/8) - 1.5^2 ] = 0.866

OR

n=3, p=.5, q=.5 Thus, µ = 1.5 and ø = 0.866

Uniform Distribution
A uniform distribution is a specific type of probability distribution that can be discrete or continuous. It is when all the outcomes occur with an equal probability. This can be shown using a table or a graph.

Example 3: This is the probability of getting a specific number when rolling a normal six-sided die.
 * Number Rolled || 1 || 2 || 3 || 4 || 5 || 6 ||
 * Probability || 1/6 || 1/6 || 1/6 || 1/6 || 1/6 || 1/6 ||

Example (Graph): It is just a straight line because the probability is the same for all outcomes. Just like all other probability distribution functions, the area between the function and the x-axis is one.

It is important to note that uniform distribution can also be continuous distribution.

Continuous Distribution
A continuous distribution gives the probability for any result between two specified values. Continuous distributions must be shown in the form of a function instead of a table. This is because there is an infinite amount of possible values for each variable, unlike discrete distribution when there is a positive integer amount of possible values. However this makes the probability of getting any specific value is zero. Continuous distributions are often used to model things with a huge number of elements like, the probability of choosing someone of a given height from everyone of a given age and gender in America. Since you could define height to the billionth of an inch, where your measuring tools are to be questioned anyway, there are so many possibilities discrete distribution doesn't make sense. For most probability distributions a probability density function is used. There are different kinds of probability density functions, but the most commonly used one is called the normal distribution. In addition, all probability distributions have an area (or integral if you're feeling fancy) of 1 between the function and the x-axis, because the probability of getting anything at all needs to be 1 or 100%.

Example: This is an example of a normal distribution. The shaded area represents the probability that a random variable x has a length within the shaded area. In general, a smaller area underneath the curve indicates a smaller probability, and a larger area indicates a larger probability. The entire area between the bell curve and the x-axis is 1 because the length must be a positive real number, and an area of 1 means a probability of 100%. Also, the probability that x is any one specific value is zero. If this is confusing to you think of the probability of a specific X as the area of a straight line, or a rectangle with a width of 0, therefor with an area of 0.

Example 2: -A continuous uniform (or rectangular) distribution looks like this. The flat top line indicates that all outcomes have an equally likely chance of occurring, while the flat bottom line indicates that all other possible outcomes have no possibility of occurring. -Finding the probability of a certain outcome is easy; you find the area of the rectangle formed by the desired interval. For example: You arrive randomly between 1:00 and 1:30. What is the probability you arrive between 1:10 and 1:15? The favorable length is 5 minutes and the total length is 30 minutes. Therefore, P(you arrive between 1:10 and 1:15) = desired outcome/total outcomes = 5 mins/30 mins = 1/6 To use a graph to solve this, you would simply find the area formed by the base (or desired time) and the height (or probability of each event occurring). The base would be 5, and the probability would be 1/30. 5*1/30 = 1/6

What's the probability you arrive at 1:15? 0, because 1:15 has length zero. Only //intervals// of time have a probability greater than 0.

1)If there is a uniform probability distribution of getting 400 - 800, then what is the possibility of getting a number from 500-1000? Ok, so probability is wanted outcomes over total outcomes.
 * Uniform Distribution Problems:**

Total outcomes = (800-400) = 400 This is because the possibility of getting a number less than 400 or greater than 800 is 0.

Wanted outcomes = wait... isn't 800 to 1000 not inside the probability distribution? YES! So the wanted outcomes is simply 800-500 (b/c anything greater than 800 has 0% chance of occurring) = 300. So the answer is 300/400 = .75

http://www.elderlab.yorku.ca/~aaron/Stats2022/BinomialDistribution.htm http://stattrek.com/probability-distributions/binomial.aspx?Tutorial=Stat https://controls.engin.umich.edu/wiki/index.php/Discrete_Distributions:_hypergeometric,_binomial,_and_poisson http://mathworld.wolfram.com/UniformDistribution.html
 * Sources:**