Probability Density Function

• It is a function whose area under the graph represents the probability used for continuous random variables
• It is represented by f(x)

• Conditions:
  • The total area is always = 1
    
  • It cannot have negative probabilities, therefore the graph cannot dip below the x-axis; f(x) ≥ 0
  • The probability that X lies between a and b is the area from a to b
    • P(a < X < b) =
    • Outside the given interval f(x) = 0
    • P(X = b) is always = 0 as there is no area
  • NOTE:
    • P(X < b) = P(X ≤ b) as no extra area is added
    • The mode of a pdf is its maximum point (stationary point)

Examples

1. Given that
   
   1. Find the value of k
      • The total area must equal 1, hence:
        • 
        • = 75k – ¹²⁵/₃ k – 12k + ⁸/₃ k = 24k = 1
        • ∴ k = ¹/₂₄
   2. Find the mode, m
      • Mode is the value which has the greatest probability, hence we are looking for the maximum point of the pdf:
        • ^d/_dx[kx(6 – x)] = 6k – 2kx
      • Finding the maximum point (stationary point):
        • 6k – 2kx = 0
        • 6k = 2kx
        • x = 3
        • ∴ mode = 3
   3. Find P(X < m)
      • P(X < m) can be interpreted as P(-∞ < X < m)
        
      • = ¹/₂₄(3(3²) – ^3³/₃ – 3(2²) + ^2³/₃) = ¹³/₃₆

Mean and Variance

• To calculate mean/expectation:
  
• To calculate variance:
  • First calculate E(X) then E(X²) by
    
  • Substitute information and calculate using
    • Var(X) = E(X²) – E(X)²

The Median

The Cumulative Distribution Function (cdf)

• It gives the probability that the value is less than b
  • P(X < b) or P(X ≤ b)
• It is represented by F(b)
• It is the integral of f(x)
  
• Median: is the value of b for which F(b) = 0.5