Probability Distributions

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Probability distributions reveal the probabilities associated with each outcome available to a random variable. These outcomes may be individual values (as with discrete probability distributions) or be a interval of values (as with continuous probability distributions).

In the realm of statistics, there are two types of probability distributions: Discrete Probability Distributions and Continuous Probability Distributions.

*Discrete Probability Distributions

A practical illustration of a discrete probability distribution can be found in rolling dice. There are only 6 possible outcomes if you roll a die: 1, 2, 3, 4, 5, and 6. The probabilities that you roll any of these outcomes happen to be the same, all 1/6, and the sum of these the probabilities of the six different outcomes is 1.

For a more mathematical explanation: A discrete probability distribution can be defined as a distribution with a finite set of countable outcomes whose sum is equal to 1. If we let x represent all the values running through the set of these outcomes while X represents the discrete random variable (that is a random outcome), then a discrete probability distribution is such that:

\sum_x^{} Pr[X = x] = 1


*Continuous Probability Distributions

On the other hand, continuous probability distribution can be illustrated with the example of measuring the weight of a bushel of apples. The outcomes that you would obtain from measuring the weight of each apple would vary like so: 150.534... grams, 149.259... grams, 154.274... grams, 152.389... grams, and so on. As you can see, there are an infinite number of outcomes that emerge from these measurements. Thus, the probability that an apple would weight exactly 152.234... grams is zero.

Mathematically, continuous probability distribution can be defined as a distribution with an infinite set of uncountable outcomes. The probability that a random outcome is equal to any real-value is essentially zero, because there are an infinite number of outcomes that are possible. If again, we let x represent all the real-number values running through the set of these outcomes while X represents the continuous random variable (that is a random outcome), then a continuous probability distribution is such that:

Pr[X = x] = 0 \,

All continuous probability distributions must have a probability density function.


Cumulative Distribution Function

Another way to define these two types of distributions is through their relationship to the cumulative distribution function.

  • F(-infinity)=0
  • F(infinity)=1
  • 0<F(y)<1
  • F(y) is always increasing, F'(y) is always positive
  • F'(y) = f(y)


Probability Density Function

A probability density function (also known as the probability distribution function) of a random variable describes the probability density of each point in the set of outcomes available to the random variable.

  • A probability distribution can only have a probability density function if and only if it is a continuous probability distribution


Examples of Probability Density Functions

The most common probability density functions are in the shape of a bell curve. These...

skewed (L,R), uniform, distributed


Properties

  • function must integrate to 1

Median

here

Mode

here

Mean (Expected Value)

Visual Example

Example with gambling...

Equation: E(X) = \int_a^b xf(x)dx