## Mean and Variance of a Linear Function of a Random Variable

The above heading sounds complicated but put simply concerns what happens to the mean of a random variable if you, say, double each value, or add 6 to each value.

Quite logically, the answer is that the mean would also double and be increased by six!

The variance is changed by the doubling but the spread of the values is unchanged by adding a constant amount.

### Mean of a Linear Function of a Random Variable

The following rules express this more formally:

 The formulae In words E(a) = a The mean value of a constant a is a. E(aX) = a E(X) If each value in a probability distribution is multiplied by a, the mean of the distribution will be multiplied by a factor of a. E(aX + b) = a E(X) + b If a constant value, b, is added to or subtracted from each value in a probability distribution, the mean of the distribution will be increased or decreased by b. where a and b are constants

Example
The expected value of a probability distribution of the random variable X is 10.
i.e. E(X) = 10

Find E(3X + 4)

E(3X + 4) =3 E(X) + 4
= 3 × 10 + 4
= 34

### Variance of a Linear Function of a Random Variable

 The formulae In words VAR(a) = 0 The variance of a constant a is 0. VAR(aX) = a2 VAR(X) If each value in a probability distribution ismultiplied by a the variance of the distribution will be multiplied by a factor of a2. Proof VAR(aX + b) = a2 VAR(X) If a constant value, b, is added to or subtracted from each value in a probability distribution, the variance of the distribution will be unchanged. where a and b are constants

Example
The variance of a probability distribution of the random variable X is 2.
i.e. VAR(X) = 2

Find VAR(3X + 4)

VAR(3X + 4) = 32 VAR(X)
= 9 x 2
= 18

### Standard Deviation of a Linear Function of a Random Variable

Because the standard deviation is the square root of the variance it follows that:

 The formulae In words SD(a) = 0 The standard deivation value of a constant a is 0. SD(aX) = a SD(X) If each value in a probability distribution is multiplied by a, the standard deviation of the distribution will be multiplied by a factor of a. SD(aX + b) = a SD(X) If a constant value, b, is added to or subtracted from each value in a probability distribution, the standard deviation of the distribution will be unchanged. where a and b are constants

Example
The standard deviation of a probability distribution of the random variable X is √2.

Find SD(3X + 4)

SD(3X + 4) =3 SD(X)
= 3 x √2
= 4.24 (to 3 sig. fig)

### Finding the Mean and Variance of a Linear Function of a Random Variable from a table

Given the following probability function of the random variable X, find E(X), VAR(X), E(3X + 4) and VAR(3X + 4).

 x 5 10 15 P(X = x) 0.2 0.5 0.3

E(X) = 5 x 0.2 + 10 x 0.5 + 15 x 0.3 = 10.5

VAR(X) = E(X2) − [E(X)]2 = 52 x 0.2 + 102 x 0.5 + 152 x 0.3 − 10.52 = 12.25

Add rows to the table to find the distribution of 3X + 4

 x 5 10 15 3x + 4 19 34 49 P(X = x) 0.2 0.5 0.3

E(3X + 4) = 19 x 0.2 + 34 x 0.5 + 49 x 0.3 = 35.5

VAR(3X + 4) = E[(3X + 4)2] + [E(3X + 4)]2 = 192 x 0.2 + 342 x 0.5 + 492 x 0.3 – 35.52 =110.25

Check

E(3X + 4) = 3 E(X) + 4 = 3 × 10.5 + 4 = 35.5 (which is equal to the value worked out from the table.)

VAR(3X + 4) = 32 VAR(X) = 9 x 12.25 = 110.25 (which is equal to the value worked out from the table.)