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Continuous Random Variables

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) – /3 – 3(2²) + /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
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