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. 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

4. 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 + 8/16
= 80/16 = 5
Var(X) = E(X²) – (E(X))²
=(4/16 +18/ / / / / /16)-25
= 440/16 – 25 = 2.5

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

8. Effect of multipliers and variance

9. 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)

10. 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