Probability Density Functions

Probability Density Functions (PDFs)

  • A probability density function is a specific type of function used to represent probabilities of outcomes in experiments.

Coin Flip Example

  • Experiment: Flipping a coin.
  • Possible outcomes: Heads or tails.
  • Probability of each outcome: 1/2 (0.5) for heads, 1/2 (0.5) for tails.
  • Graphical Representation:
    • Two rectangles, each with a height of 1/2, representing the probability of heads and tails.
    • Each rectangle is one unit wide.
    • Area of each rectangle: 1 \times \frac{1}{2} = \frac{1}{2}
    • Total area: \frac{1}{2} + \frac{1}{2} = 1 (representing 100% probability)

Flipping Two Coins (Binomial Distribution)

  • Possible outcomes: 0 heads, 1 head, or 2 heads.
  • Probabilities:
    • 0 heads: 1/4
    • 1 head: 1/2
    • 2 heads: 1/4
  • Reasoning:
    • 0 heads (TT): Tails must occur twice in a row.
    • 2 heads (HH): Heads must occur twice in a row.
    • 1 head (HT or TH): Can occur in two ways (head then tail, or tail then head), making it twice as likely as 0 or 2 heads.
  • Total probability: \frac{1}{4} + \frac{1}{2} + \frac{1}{4} = 1

Normal Curve

  • The normal curve is an example of a PDF.
  • Total area under the curve = 1.
  • Area between one standard deviation to the left and right of the mean is approximately 0.68.

Experiment with Four-Faced Dice

  • Scenario: Using three four-faced dice (pyramids with faces numbered 1 to 4).
  • Method: Roll the dice and sum the numbers on the bottom faces.
  • Possible outcomes:
    • Minimum sum: 3 (1+1+1)
    • Maximum sum: 12 (4+4+4)
    • Possible sums: 3, 4, 5, …, 12

Simulation Using a Calculator/Computer

  • Objective: Simulate the experiment to estimate probabilities.
  • Lists:
    • Die 1: Random integers from 1 to 4 (1000 rolls).
    • Die 2: Random integers from 1 to 4 (1000 rolls).
    • Die 3: Random integers from 1 to 4 (1000 rolls).
    • Total: Sum of the three dice for each roll.
  • Using relative proportions from the simulation to approximate theoretical probabilities.

Histogram Representation

  • X-axis: Possible totals (3 to 12).
  • Y-axis: Frequency or percentage of each total.
  • Each rectangle is one unit wide.
  • Example interpretation: If 2% of the simulations resulted in a total of 3, the rectangle above '3' would have a height of 2%.
  • The sum of all percentage heights equals 100%.

Real-World Application: Survey on Number of Televisions per Family

  • Method: Conduct a survey using random sampling.
  • Data collection: Ask families how many televisions they own.
  • Data representation: Create a PDF with:
    • X-axis: Number of televisions.
    • Y-axis: Percentage of families with that number of televisions.

Key Characteristic of a PDF

  • A PDF is a graph with numbers on the x-axis (representing possible outcomes) and an area above the x-axis equal to exactly one (representing 100% probability).

Formula for the Area of PDF

Area = 1

  • The area captures total probability, ensuring every possibility is accounted for.