Statistics 1 Revision Notes - Mr Barton Maths

Example:
score on two dice, X 2 3 4 5 6 7 8 9 10 11 12
probability, f (x)
is the probability distribution for the random variable, X, the total score on two dice.
Note that the sum of the probabilities must be 1, i.e. P ( X = x)
= 1.
12
∑
x= 2
Cumulative probability distribution
Just like cumulative frequencies, the cumulative probability, F, that the total score on two dice
is less than or equal to 4 is F(4) = P(X ≤ 4) = .
Note that F(4.3) means P(X ≤ 4.3) and seeing as there are no scores between 4 and 4.3 this is
the same as P(X ≤ 4) = F(4).
Expectation or expected values
Expected mean or expected value of X.
For a discrete probability distribution the expected mean of X , or the expected value of X is
μ = E[X] =
Expected value of a function
The expected value of any function, f (X), is defined as
E[X] =
Note that for any constant, k, E[k] = k,
since ∑ = k ∑ = k × 1 = k
Expected variance
The expected variance of X is
Var ,
or
σ 2 = Var[X] = E[(X – μ) 2 ] = E[X 2 ] – μ 2 = E[X 2 ] – (E[X]) 2
34 14/04/2013 Statistics 1 SDB