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

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

Variance Var(X) = E(X²) – (E(X))² a) x 2 3 4 5 6 7 8 p(x) 1/16 2/16 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) + 1 2 3 4 5 6 7 8 x 2 3 4 5 6 7 8 p(x) 1/16 2/16 3/16 4/16

Variance Var(X) = E(X²) – (E(X))² x 2 3 4 5 6 7 8 p(x) 1/16 2/16 3/16 b) Find E(M) x 2 3 4 5 6 7 8 p(x) 1/16 2/16 3/16 4/16 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

The random variable X has probability function P(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)

Effect of multipliers and variance

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

Effect of multiplier and constant on E(X) and Var(X) 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