### 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
•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
Example
2 four sided die numbered 1,2,3,4 are spun and their faces are
a) Find the probability distribution of X
b) 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
2
3
x
p(x)
1/
16
2/
16
4
3/
16
5
4/
16
6
3/
16
7
2/
16
8
1/
16
Variance
Var(X) = E(X²) – (E(X))²
b) Find E(M)
x
p(x)
2
1/
16
3
2/
16
4
3/
16
5
4/
16
6
3/
16
7
2/
16
8
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 function
P(X = x) = kx,
x = 1,2,3
k(x+1)
x = 4,5
where k is a constant.
(a)
(b)
(c)
(d)
Find the value of k.
(2)
Find the exact value of E(X).
(2)
Show that, to 3 significant figures, Var(X) = 1.47. (4)
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
```