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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

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Expected value and variance formulae E(X) = ΣxP(X=x) = Σxp(x) E(X²) = Σx²p(x) E(X n ) = Σx n p(x) Var(X) = E(X²) – (E(X))²

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Variance Var(X) = E(X²) – (E(X))² Example 2 four sided die numbered 1,2,3,4 are spun and their faces are added (X). a)Find the probability distribution of X b)Find E(M) c)Find Var(M) a) x p(x) 1 / 16 2 / 16 3 / 16 4 / 16 3 / 16 2 / 16 1 / 16

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Variance Var(X) = E(X²) – (E(X))² b) Find E(M) x p(x) 1 / 16 2 / 16 3 / 16 4 / 16 3 / 16 2 / 16 1 / 16 E(M) = Σxp(x) = 2 / / / / / / / 16 = 80 / 16 = 5 Var(X) = E(X²) – (E(X))² =( 4 / / / / / / / 16 )-25 = 440 / 16 – 25 = 2.5

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The random variable X has probability function P(X = x) = kx, x = 1,2,3 k(x+1) x = 4,5where 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) = (4) (d)Find, to 1 decimal place, Var(4 – 3X). (2) (Total 10 marks)

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Effect of multipliers and variance

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Effect of multiplier and constant on E(X) and Var(X) E(X)= 3 and Var(X)=5 a)Calculate E(2X) b)Calculate E(X+6) c)Find Var(3X) d)Find E(4X-1) e)Find Var(4X-1) f)Find Var(2-3X)

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Effect of multiplier and constant on E(X) and Var(X) E(X)= 3 and Var(X)=5 a)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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