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Cumulative Distribution Function in Probability Theory

Learn how to use the cumulative distribution function to calculate probabilities for random variables with examples and exercises.

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Cumulative Distribution Function in Probability Theory

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  1. F(x) 1 x 0 a b Cumulative Distribution Function F(x) It is also possible to use the cumulative probability function to calculate probability. F(x) = P(X ≤ x) for all x. The probability is 0 for any value under a, and 1 for any value over b. F(x) = 0, for x < a F(b) = 1, for x > b

  2. d c c To find P(c ≤ x ≤ d) F(x) x b a To find P(x ≤ c) F(x) P(x ≤ c) = F(c) x b a P(c ≤ x ≤ d) = F(d) – F(c)

  3. Example • The continuous random variable X is distributed with cumulative distribution function F where • F(x) = 0 for x < 1 • F(x) = for 1 ≤ x ≤ 3 • F(x) = 1 for x > 3 • Find • P(X < 2) • P(X > 1½) • P(1½ < X ≤ 2½)

  4. P(X < 2) = F(2) = b) P(X > 1½) = 1 – F(1½) = c) P(1½ < X ≤ 2½) = F(2½) - F(1½)

  5. Change Probability Density Function f(x) to Cumulative Distribution Function F(x) f(x)  F(x) Since F(x) = P(X ≤x),in order to change f(x) to F(x) we must integrate between the lower limit and x. for 0 ≤ x ≤ 4

  6. F(x) = 0 for x < 0 F(x) = for 0 ≤ x ≤ 4 F(x) = 1 for x > 4 Cumulative Distribution Function Exercise

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