PROBABILITY

1. The number of heads obtained when flipping 3 coins is the discrete random variable, X which has the following probability distribution.

x	0	1	2	3
P(X = x)	1/8	3/8	3/8	1/8

Calculate E(X)

Calculate the expectation and variance of X of the following distribution:

x	0	1	2	3
P(X = x)	0.1	0.2	0.5	0.2

We already know how to calculate E(X) and E(X²).

E(X) = (0 × 0.1) + (1 × 0.2) + (2 × 0.5) + (3 × 0.2) = 1.8

E(X²) = (0 × 0.1) + (1 × 0.2) + (4 × 0.5) + (9 × 0.2) = 4

So:

Var(X) = E(X²) − (E(X))² = 4 − 1.8² = 0.76

There are a few general results we should remember to help with our calculations...

Var(aX) = a²Var(X)

Var(aX + b) = a²Var(X)

Where a and b are both constants.

This means by knowing just the variance, Var(X), we can calculate other variances quickly.

Example:

If Var(X) = 2.5 then,

Var(2X) = 2² × Var(X) = 4 × 2.5 = 10

Var(4X − 3) = 4² × Var(X) = 16 × 2.5 = 40

Read more at http://www.s-cool.co.uk/a-level/maths/probability-distributions/revise-it/expectation-and-variance#81dm37Ku2li51jmH.99
Read more at http://www.s-cool.co.uk/a-level/maths/probability-distributions/revise-it/expectation-and-variance#81dm37Ku2li51jmH.99

We can work out the expectation as follows...

E(X) = (0 × 1/8) + (1 × 3/8) + (2 × 3/8) + (3 × 1/8) = 1.5

Read more at http://www.s-cool.co.uk/a-level/maths/probability-distributions/revise-it/expectation-and-variance#81dm37Ku2li51jmH.99
Notes available at
http://www.s-cool.co.uk/a-level/maths/probability-distributions/revise-it/expectation-and-variance

Integration

A straight line with gradient 3 passes through the point  (1,4). Find the equation of the line.

dy/dx=3
dy=3dx

∫dy=∫3dx
y=3x+c
substitute 1,4 and findout c
4=3+c
c=4-3=1
So the equation is y=3x+1