Probability Functions - Discrete & Continuous

Random Variables

• A random variable is one whose values depend on the outcomes of a random phenomenon.

• A random variable can be discrete or continuous.

Discrete Probability Distributions

• A probability distribution is a list of all possible outcomes of a random variable along with their corresponding probability values.

• Example: Outcome of a roll of a die:

  • Outcome 1: Probability 1/6

  • Outcome 2: Probability 1/6

  • Outcome 3: Probability 1/6

  • Outcome 4: Probability 1/6

  • Outcome 5: Probability 1/6

  • Outcome 6: Probability 1/6

Probability Functions

• Functions describe the relation between an input variable and an output variable.

• We can find the value of the outcome variable, given the value of the input variable.

• $f(x) = x + 7$ (Example function)

• Variables are represented by uppercase letters.

• Values of a variable are represented by lowercase letters.

Parameters

• Parameter defines the function.

• Usually written in Greek letters (e.g., β).

Probability Mass Function (PMF)

• $f(x) = P(X = x)$

• Because it outputs a probability P.

• In a PMF:

  • Output values are between 0 and 1.

  • The sum of all outcomes is equal to 1.

• A probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.

Probability Density Function (PDF)

• Deals with continuous random variables.

• Unlike PMF, gives the probability of obtaining a specific value.

• Example: Hedgehog Height
  Bin Width = 4
  *Density = Frequency / Bin Width
  *Relative Density = Density / cf , where cf is the frequency

• Example Calculation: Density = $37 / 4 = 9.25$

• Example Calculation: Relative density = $9.25 / 196 = 0.047$

• Area = Relative density * Bin Width. The bin height is the probability P of getting a hedgehog height (h) between 13 and 16.99 cm.

• Area = $0.047 * 4 = 0.188$

• In a PDF:

  • Output values are greater than or equal to 0.

  • The output values of a PDF are not probabilities.

  • The probabilities of obtaining a given value are estimated by the area under that curve or line.

Key Differences

• Discrete probability functions are referred to as probability mass functions (PMF).

• Continuous probability functions are referred to as probability density functions (PDF).

Summary

• A random variable is any variable whose value is the outcome of a random event.

• A probability distribution is a list of all the possible values a random variable can take along with their corresponding probability.

• With the function that describes a probability distribution, we can easily calculate the probability of any outcome for a random variable.

• For discrete random variables, we use a probability mass function or PMF.

• For continuous random variables, we use a probability density function or PDF.

• Unlike the PMF, in a PDF it is the area between two values or the area beyond or before one value that will tell us the probability of obtaining a specified value for the random variable.

• The whole area under the curve of a PDF will sum up to 1.