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Discrete Random Variables 3

Discrete Random Variables 3. To be able to calculate the expected value and variance of a discrete random variable To investigate the effect of multipliers and constants on the expected value and the variance of a discrete random variable

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Discrete Random Variables 3

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  1. Discrete Random Variables 3 • To be able to calculate the expected value and variance of a discrete random variable • To investigate the effect of multipliers and constants on the expected value and the variance of a discrete random variable • To be able to calculate the expected value and variance of distributions like y=aX+b

  2. Expected value and variance formulae E(X) = ΣxP(X=x) = Σxp(x) E(X²) = Σx²p(x) E(Xn) = Σxnp(x) Var(X) = E(X²) – (E(X))²

  3. Var(X) = E(X²) – (E(X))² Variance • Example • 2 four sided die numbered 1,2,3,4 are spun and their faces are added (X). • Find the probability distribution of X • Find E(M) • Find Var(M) a)

  4. Var(X) = E(X²) – (E(X))² Variance b) Find E(M) E(M) = Σxp(x) =2/16 +6/16 +12/16 +20/16 +18/16 +14/16 +8/16 = 80/16 = 5 Var(X) = E(X²) – (E(X))² =(4/16 +18/16 +48/16 +100/16 +108/16 +98/16 +64/16)-25 = 440/16 – 25 = 2.5

  5. The random variable X has probability functionP(X = x) = kx, x = 1,2,3 k(x+1) x = 4,5 where k is a constant.(a) Find the value of k. (2)(b) Find the exact value of E(X). (2)(c) Show that, to 3 significant figures, Var(X) = 1.47. (4)(d) Find, to 1 decimal place, Var(4 – 3X). (2) (Total 10 marks)

  6. Effect of multipliers and variance

  7. Effect of multiplier and constant on E(X) and Var(X) • E(X)= 3 and Var(X)=5 • Calculate E(2X) • Calculate E(X+6) • Find Var(3X) • Find E(4X-1) • Find Var(4X-1) • Find Var(2-3X)

  8. Effect of multiplier and constant on E(X) and Var(X) • E(X)= 3 and Var(X)=5 • E(2X) = 2E(X) = 2 x 3 = 6 b) E(X+6) = E(X)+6 = 3+6 = 9 c) Var(3X) = 3²Var(X) = 9x5 = 45 d) E(4X-1) = 4E(X)-1 = 4x3-1 = 11 e) Var(4X-1) = 4²Var(X) = 16x5 = 80 f) Var(2-3X) = -3²Var(X) = 9x5 = 45

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