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))²Example2 four sided die numbered 1,2,3,4 are spun and their faces are

added (X).a) Find the probability distribution of Xb) Find E(M)c) Find Var(M)

a) + 1 2 3 4

1 2 3 4 5

2 3 4 5 6

3 4 5 6 7

4 5 6 7 8

x 2 3 4 5 6 7 8

p(x) 1/162/16

3/164/16

3/162/16

1/16

Variance Var(X) = E(X²) – (E(X))²

b) Find E(M)

x 2 3 4 5 6 7 8

p(x) 1/162/16

3/164/16

3/162/16

1/16

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 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)

Effect of multipliers and variance

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)

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