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E(X 2 ) =

E(X 2 ) =. E(X) =. The Mean and Variance of a Continuous Random Variable. In order to calculate the mean or expected value of a continuous random variable, we must multiply the probability density function f(x) with x before we integrate within the limits.

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E(X 2 ) =

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  1. E(X2) = E(X) = The Mean and Variance of a Continuous Random Variable In order to calculate the mean or expected value of a continuous random variable, we must multiply the probability density function f(x) with x before we integrate within the limits. To calculate the variance, we need to find E(X2) since Var (X) = E(X2) – [E(X)]2

  2. Example • The continuous random variable X is distributed with probability density function f(x) where • f(x) = 6x(1-x) ar gyfer 0 ≤ x ≤ 1 • a) Calculate the mean and variance of X. • b) Deduce the mean and variance of • Y = 10X – 3 • Z = 2(3 – X) • 5 • Evaluate E(5X2 – 3X + 1)

  3. E(X) = • Calculate the mean and variance of X. f(x) = 6x(1-x) = 6x – 6x2

  4. Var (X) = Var (X) = E(X2) – [E(X)]2 E(X2) =

  5. 10 x 1 – 3 = 2 6 – 2 x 1 = 5 5 2 22x 1 = 5 20 22x Var (X) = 5 • Deduce the mean and variance of • (i) Y = 10X – 3 • (ii) Z = 2(3 – X) • 5 (i) E(Y) = E(10X – 3) = 10E(X) – 3 = 2 Var(Y) = Var(10X – 3) = 102 Var(X) = 100 x 1 = 20 5 (ii) E(Z) = E 6 – 2X = 5 5 6 – 2E(X) = 5 5 1 Var(Z) = Var 6 – 2X = 5 5 1 . 125

  6. 5 x 3 - 3 x 1 + 1 = 10 2 c) Evaluate E(5X2 – 3X + 1) E(5X2 – 3X + 1) = 5E(X2) – 3E(X) + 1 = 1 Exercise 1.4 Mathematics Statistics Unit S2 - WJEC Homework 11 Homework 12

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