Expectation+Probability+AS+3.3


 * Variance of a random variable.**

The ** standard deviation **is a **measure of spread.** However most results concerning the standard deviation are proved using the ** variance, **
 * which ** **is the square of the standard deviation.**
 * we use the symbol, ** σ **, for the ** standard deviation or SD(X) **
 * we use the symbol**,** **σ2****, ** for the variance, or **VAR(X)**

A useful guide as to when to use each Use Greek symbols, µ, σ, σ2, when they are given, or already calculated Use ** E(X), SD(X), VAR(X) ,** in proofs or when they have ** to be worked out .**

The definition of variance is: This is the formula you used in yr 12
 * VAR(X) = E(X - µ)2**

But is general practice it is much easier to use this form of the formula. A proof can be used to establish they are equal.


 * VAR(X) = E(X2) - µ2 **
 * OR VAR(X) = E(X2) – [E(X)]2 **

Calculate the variance and standard deviation of X.
 * Example.**


 * x || 2  ||  7  ||
 * P(X = x || 0.2  ||  0.8  ||

E(X2) = 22 x 0.2 + 72 x 0.8 = 0.8 + 39.2 = 40

E(X) = µ = 2 x 0.2 + 7 x 0.8 = 6 [E(X)]2 = 36 VAR(X) = 40 – 36 = 4 SD(X) = 2


 * Mean of a function of a random variable. **

Any linear function can be written in the form aX + b What is the expected value **of aX + b?**

A motel unit sleeps a maximum of four people. When it is occupied, the number of people is given by the probability distribution
 * Demonstration example. **


 * n || 1  ||  2  ||  3  ||  4  ||
 * P(N = n) || 0.2  ||  0.4  ||  0.3  ||  0.1  ||

E(N) =2.3 This means, “on average”, each unit is occupied by 2.3 people.

The tariff is $20 per person. Calculate the expected (mean) tariff.

The table giving the probability distribution of the tariff is:


 * t || 20  ||  40  ||  60  ||  80  ||
 * P(T = t) || 0.2  ||  0.4  ||  0.3  ||  0.1  ||

E(T) = 20 x 0.2 + 40 x 0.4 + 60 x 0.3 + 80 x 0.1 = 46

Note that T = 20N, and this result could have been calculated directly

More realistically the charge per night would be $20 per person and $30 for the use of the unit. Calculate the mean for the income for the unit.


 * i || 50  ||  70  ||  90  ||  110  ||
 * P(I = i) || 0.2  ||  0.4  ||  0.3  ||  0.1  ||

E(I) = 50 x 0.2 + 70 x 0.4 + 90 x 0.3 + 110 x 0.1 = 76 Note that I = T + 30 and there is also a shortcut available.

E(I) = E( 20N + 30) = 20 x E(N) + 30 = 20 x 2.3 + 30 = 76

Summary:
 * E(aX) = a x E(X) **
 * E(X + b) = E(X) + b **
 * E(aX + b) = a x E(X) + b these results are valid only for linear functions of a random variable **


 * BUT NOTE!!!!**

The process above is only valid for **linear functions of a random variable**, in particular, E(X2) is not equal to [E(X)]2 or E(X2 ) is not equal to µ2

Example:


 * //x// || 2  ||  7  ||
 * P(X = //x//) || 0.2  ||  0.8  ||

E(X) = 2 x 0.2 + 7 x 0.8 = 6 E(X2) = 22 x 0.2 + 72 x 0.8 = 4 x 0.2 + 49 x 0.8 = 40 **→ E(X2 ) is not equal to [E(X)]2**